What is AVL Tree?
A Self-balancing Binary Search Tree (BST) where difference between heights of left and right subtrees cannot be more than 1 for all nodes is called an AVL Tree.
Below is a simple implementation of AVL tree in C++ with in-line explanation in comments.
Full Code:
<![CDATA[#include <iostream>
#include <ctime>
#include <iomanip>
// **************************************************
// Adelson-Velskii and Landis dynamically balanced
// tree class.
class AVLTree {
// - - - - - - - - - - - - - - - - - - - - - - - -
// Private Node subclass for the nodes of the tree.
private:
class Node {
// - - - - - - - - - - - - - - - - - - - - - -
// Private data for the node of the tree.
private:
// The actual data being stored.
int data;
// The height of this node from the deepest
// point.
int height;
// A pointer to the left subtree.
Node *left_child;
// A pointer to the node's parent.
Node *parent;
// A pointer to the right subtree.
Node *right_child;
// - - - - - - - - - - - - - - - - - - - - - -
// Public methods to process the node.
public:
// Constructor initializing the data.
Node(int val);
// Calculate the balance point.
int getBalance();
// Get the actual data.
int getData();
// Get the height.
int getHeight();
// Get the left subtree.
Node *getLeftChild();
// Get the node's parent.
Node *getParent();
// Get the right subtree.
Node *getRightChild();
// Remove the node's parent.
void removeParent();
// Set the left subtree.
Node *setLeftChild(Node *newLeft);
// Set the right subtree.
Node *setRightChild(Node *newRight);
// Set the node's height.
int updateHeight();
}; // Node
// - - - - - - - - - - - - - - - - - - - - - - - -
// Private data for the tree.
private:
// A pointer to the top node of the tree.
Node *root;
// - - - - - - - - - - - - - - - - - - - - - - - -
// Private methods to process the tree.
private:
// Balance the subtree.
void balanceAtNode(Node *n);
// Find the node containing the data.
Node *findNode(int val);
// Print the subtree.
void printSubtree(Node *subtree, int depth,
int offset, bool first);
// Rotate the subtree left.
void rotateLeft(Node *n);
// Rotate the subtree left.
void rotateRight(Node *n);
// Set the root.
void setRoot(Node *n);
// Figure out the default spacing for element.
int spacing(int offset);
// - - - - - - - - - - - - - - - - - - - - - - - -
// Public methods to process the tree.
public:
// Constructor to create an empty tree.
AVLTree();
// Constructor to populate the tree with one node.
AVLTree(int val);
// Get the tree's height.
int getHeight();
// Insert the value into the tree.
bool insert(int val);
// Print the tree.
void print();
// Remove the value from the tree.
bool remove(int val);
}; // AVLTree
// **************************************************
// AVLTree's Node subclass methods.
// --------------------------------------------------
// Constructor initializing the data.
AVLTree::Node::Node(int val) {
data = val;
height = 0;
parent = nullptr;
left_child = nullptr;
right_child = nullptr;
} // Node
// --------------------------------------------------
// Calculate the balance point. Negative, when the
// right side is deeper. Zero, when both sides are
// the same. Positive, when the left side is deeper.
int AVLTree::Node::getBalance() {
// If we don't have a left subtree, check the
// right.
int result;
if (left_child == nullptr)
// If neither subtree exists, return zero.
if (right_child == nullptr)
result = 0;
// Otherwise, the right subtree exists so make
// it's height negative and subtract one.
else
result = -right_child->height-1;
// Otherwise, we have a left subtree so if we
// don't have a right one, return the left's
// height plus one.
else if (right_child == nullptr)
result = left_child->height+1;
// Otherwise, both subtrees exist so subtract
// them to return the difference.
else
result = left_child->height-right_child->height;
return result;
} // getBalance
// --------------------------------------------------
// Get the actual data.
int AVLTree::Node::getData() {
return data;
} // getData
// --------------------------------------------------
// Get the height.
int AVLTree::Node::getHeight() {
return height;
} // getHeight
// --------------------------------------------------
// Get the left subtree.
AVLTree::Node *AVLTree::Node::getLeftChild() {
return left_child;
} // getLeftChild
// --------------------------------------------------
// Get the node's parent.
AVLTree::Node *AVLTree::Node::getParent() {
return parent;
} // getParent
// --------------------------------------------------
// Get the right subtree.
AVLTree::Node *AVLTree::Node::getRightChild() {
return right_child;
} // getRightChild
// --------------------------------------------------
// Remove the node's parent.
void AVLTree::Node::removeParent() {
parent = nullptr;
} // removeParent
// --------------------------------------------------
// Set the left subtree.
AVLTree::Node *AVLTree::Node::setLeftChild(
Node *newLeft) {
// If there is a left node, set it's parent to
// be us. In any event, make it our left subtree
// and update the height.
if (newLeft != nullptr)
newLeft->parent = this;
left_child = newLeft;
updateHeight();
return left_child;
} // setLeftChild
// --------------------------------------------------
// Set the right subtree.
AVLTree::Node *AVLTree::Node::setRightChild(
Node *newRight) {
// If there is a right node, set it's parent to
// be us. In any event, make it our right subtree
// and update the height.
if (newRight != nullptr)
newRight->parent = this;
right_child = newRight;
updateHeight();
return right_child;
} // setRightChild
// --------------------------------------------------
// Set the node's height.
int AVLTree::Node::updateHeight() {
// If we don't have a left subtree, check the
// right.
if (left_child == nullptr)
// If we don't have either subtree, the height
// is zero.
if (right_child == nullptr)
height = 0;
// Otherwise, we only have the right subtree so
// our height is one more than it's height.
else
height = right_child->height+1;
// Otherwise, the left subtree exists so if we
// don't have a right one, our height is one
// more than it's height.
else if (right_child == nullptr)
height = left_child->height+1;
// Otherwise, we have both subtree's so if the
// left's height is greater than the right, our
// height is one more than it.
else if (left_child->height > right_child->height)
height = left_child->height+1;
// Otherwise, the right subtree's height will be
// used so our height is one more than it.
else
height = right_child->height+1;
return height;
} // updateHeight
// **************************************************
// AVLTree class methods.
// --------------------------------------------------
// Constructor to create an empty tree.
AVLTree::AVLTree() {
root = nullptr;
} // AVLTree
// --------------------------------------------------
// Constructor to populate the tree with one node.
AVLTree::AVLTree(int val) {
root = new Node(val);
} // AVLTree
// --------------------------------------------------
// Balance the subtree.
void AVLTree::balanceAtNode(Node *n) {
// Get the current balance and if it is bad
// enough on the left side, rotate it right
// adjusting the subtree left, if required.
int bal = n->getBalance();
if (bal > 1) {
if (n->getLeftChild()->getBalance() < 0)
rotateLeft(n->getLeftChild());
rotateRight(n);
// Otherwise, if the balance is bad enough on
// the right side, rotate it left adjusting the
// subtree right, if required.
} else if (bal < -1) {
if (n->getRightChild()->getBalance() > 0)
rotateRight(n->getRightChild());
rotateLeft(n);
} // if
} // balanceAtNode
// --------------------------------------------------
// Find the node containing the data.
AVLTree::Node *AVLTree::findNode(int val) {
// While nodes remain, if we found the right
// node, exit the loop. If the value we want
// is less than the current, check the left
// subtree, otherwise, the right.
Node *temp = root;
while (temp != nullptr) {
if (val == temp->getData())
break;
else if (val < temp->getData())
temp = temp->getLeftChild();
else
temp = temp->getRightChild();
} // while
return temp;
} // findNode
// --------------------------------------------------
// Get the tree's height.
int AVLTree::getHeight() {
return root->getHeight();
} // getHeight
// --------------------------------------------------
// Insert the value into the tree.
// Returns:
// true: If addition is successful
// false: If item already exists
//
bool AVLTree::insert(int val) {
// If the tree is empty, add the new node as the
// root.
if (root == nullptr)
root = new Node(val);
// Otherwise, we need to find the insertion point.
else {
// Starting at the tree root search for the
// insertion point, until we have added the
// new node.
Node *added_node = nullptr;
Node *temp = root;
while (temp != nullptr && added_node == nullptr) {
// If the value is less than the current
// node's value, go left. If there isn't a
// left subtree, insert the node, otherwise,
// it is next to check.
if (val < temp->getData()) {
if (temp->getLeftChild() == nullptr) {
added_node = temp->setLeftChild(
new Node(val));
} else
temp = temp->getLeftChild();
// Otherwise, if the value is greater than
// the current node's value, go right. If
// there isn't a right subtree, insert the
// node, otherwise, it is next to check.
} else if (val > temp->getData()) {
if (temp->getRightChild() == nullptr) {
added_node = temp->setRightChild(
new Node(val));
} else
temp = temp->getRightChild();
// Otherwise, the value is already in the
// tree so abort.
} else
return false;
} // while
// From the new node upwards to the root,
// updated the height and make sure the
// subtree is balanced.
temp = added_node;
while(temp != nullptr) {
temp->updateHeight();
balanceAtNode(temp);
temp = temp->getParent();
} // while
} // if
return true;
} // insert
// --------------------------------------------------
// Print the tree in this pattern complaining if
// too deep or empty.
// 08
// 04 12
// 02 06 10 14
//01 03 05 07 09 11 13 15
void AVLTree::print() {
// If the tree is empty, say so.
if (root == nullptr)
std::cout << "Tree is empty!" <<
std::endl;
// Otherwise, if the tree has a height more than 4
// (5 rows), it is too wide.
else if (root->getHeight() > 4)
std::cout << "Not currently supported!" <<
std::endl;
// Otherwise, set up to display the tree. Get
// the maximum depth and for each possible
// level, output that level's elements and
// finish off the line.
else {
int max = root->getHeight();
for (int depth = 0; depth <= max; depth++) {
printSubtree(root, depth, max-depth+1, true);
std::cout << std::endl;
} // for
} // if
} // print
// --------------------------------------------------
// Print the subtree. The leftmost branch will have
// first true. The level counts up from the bottom
// for the line we are doing. The depth is how
// many layers to skip over.
void AVLTree::printSubtree(Node *subtree, int depth,
int level, bool first) {
// If we need to go deeper in the subtree, do so.
// If the subtree is empty, pass it down, otherwise
// pass both left and right subtrees.
if (depth > 0) {
if (subtree == nullptr) {
printSubtree(subtree, depth-1, level, first);
printSubtree(subtree, depth-1, level, false);
} else {
printSubtree(subtree->getLeftChild(), depth-1,
level, first);
printSubtree(subtree->getRightChild(), depth-1,
level, false);
} // if
// Otherwise, if the subtree is empty, display
// an empty element. The leftmost element only
// needs half the spacing.
} else if (subtree == nullptr)
std::cout << std::setw((first)?
spacing(level)/2:spacing(level)) << "-";
// Otherwise, we have a live element so display
// it. Once more, use half spacing for the
// leftmost element.
else
std::cout << std::setw((first)?
spacing(level)/2:spacing(level)) <<
subtree->getData();
} // printSubtree
// --------------------------------------------------
// Remove the value from the tree.
// Returns:
// true: If removal is successful
// false: If item is not found in the tree
//
bool AVLTree::remove(int val) {
// Find the node to delete and if none, exit.
Node *toBeRemoved = findNode(val);
if (toBeRemoved == nullptr)
return false;
// Get the parent and set the side the node is
// on of the parent.
enum {left, right} side;
Node *p = toBeRemoved->getParent();
if (p != nullptr &&
p->getLeftChild() == toBeRemoved)
side = left;
else
side = right;
// If the node to be removed doesn't have a left
// subtree, check it's right subtree to figure
// out our next move.
if (toBeRemoved->getLeftChild() == nullptr)
// If we don't have any subtrees, we are the
// leaf so our parent doesn't need us.
if (toBeRemoved->getRightChild() == nullptr) {
// If we don't have a parent, the tree is now
// empty so change the root to null and delete
// our node.
if (p == nullptr) {
setRoot(nullptr);
delete toBeRemoved;
// Otherwise, change the parent so it doesn't
// point to us, delete ourself, update the
// parent's height, and rebalance the tree.
} else {
if (side == left)
p->setLeftChild(nullptr);
else
p->setRightChild(nullptr);
delete toBeRemoved;
p->updateHeight();
balanceAtNode(p);
} // if
// Otherwise, there is a right subtree so use
// it to replace ourself.
} else {
// If we don't have a parent, the tree is now
// the right subtree and delete our node.
if (p == nullptr) {
setRoot(toBeRemoved->getRightChild());
delete toBeRemoved;
// Otherwise, change the parent so it doesn't
// point to us, delete ourself, update the
// parent's height, and rebalance the tree.
} else {
if (side == left)
p->setLeftChild(toBeRemoved->
getRightChild());
else
p->setRightChild(toBeRemoved->
getRightChild());
delete toBeRemoved;
p->updateHeight();
balanceAtNode(p);
} // if
} // if
// Otherwise, we have a left subtree so check the
// right one of the node being removed to decide
// what is next. If there isn't a right subtree,
// the left one will replace ourself.
else if (toBeRemoved->getRightChild() ==
nullptr) {
// If we don't have a parent, the tree is now
// the left subtree and delete our node.
if (p == nullptr) {
setRoot(toBeRemoved->getLeftChild());
delete toBeRemoved;
// Otherwise, change the parent so it doesn't
// point to us, delete ourself, update the
// parent's height, and rebalance the tree.
} else {
if(side == left)
p->setLeftChild(toBeRemoved->
getLeftChild());
else
p->setRightChild(toBeRemoved->
getLeftChild());
delete toBeRemoved;
p->updateHeight();
balanceAtNode(p);
} // if
// Otherwise, the node to remove has both subtrees
// so decide the best method to remove it.
} else {
// Check the balance to see which way to go.
// If the left side is deeper, modify it.
Node *replacement;
Node *replacement_parent;
Node *temp_node;
int bal = toBeRemoved->getBalance();
if (bal > 0) {
// If the right subtree of the node's
// left subtree is empty, we can move the
// node's right subtree there.
if (toBeRemoved->getLeftChild()->
getRightChild() == nullptr) {
replacement = toBeRemoved->getLeftChild();
replacement->setRightChild(
toBeRemoved->getRightChild());
temp_node = replacement;
// Otherwise, find the right most empty subtree
// of our node's left subtree and it's
// parent. This is our replacement. Make it's
// parent point to it's left child instead
// of itself. It is now free to replace the
// deleted node. Copy both of the deleted
// nodes subtrees into the replacement leaving
// fixing up the parent below.
} else {
replacement = toBeRemoved->
getLeftChild()->getRightChild();
while (replacement->getRightChild() !=
nullptr)
replacement = replacement->getRightChild();
replacement_parent = replacement->getParent();
replacement_parent->setRightChild(
replacement->getLeftChild());
temp_node = replacement_parent;
replacement->setLeftChild(
toBeRemoved->getLeftChild());
replacement->setRightChild(
toBeRemoved->getRightChild());
} // if
// Otherwise, we are going to modify the right
// side so, if the left subtree of the node's
// right subtree is empty, we can move the
// node's left subtree there.
} else if (toBeRemoved->getRightChild()->
getLeftChild() == nullptr) {
replacement = toBeRemoved->getRightChild();
replacement->setLeftChild(
toBeRemoved->getLeftChild());
temp_node = replacement;
// Otherwise, find the left most empty subtree
// of our node's right subtree and it's
// parent. This is our replacement. Make it's
// parent point to it's right child instead
// of itself. It is now free to replace the
// deleted node. Copy both of the deleted
// nodes subtrees into the replacement leaving
// fixing up the parent below.
} else {
replacement = toBeRemoved->
getRightChild()->getLeftChild();
while (replacement->getLeftChild() !=
nullptr)
replacement = replacement->getLeftChild();
replacement_parent = replacement->getParent();
replacement_parent->setLeftChild(
replacement->getRightChild());
temp_node = replacement_parent;
replacement->setLeftChild(
toBeRemoved->getLeftChild());
replacement->setRightChild(
toBeRemoved->getRightChild());
} // if
// Fix the parent to point to the new root.
// If there isn't a parent, update the actual
// tree root. Otherwise, there is a parent so
// if we were the left subtree, update it,
// otherwise, the right. In all cases, delete
// the node and rebalance the tree.
if (p == nullptr)
setRoot(replacement);
else if (side == left)
p->setLeftChild(replacement);
else
p->setRightChild(replacement);
delete toBeRemoved;
balanceAtNode(temp_node);
} // if
return true;
} // remove
// --------------------------------------------------
// Rotate the subtree left.
void AVLTree::rotateLeft(Node *n) {
// Get the node's parent and if it exists and the
// node was it's left subtree, remember we are
// processing the left, otherwise, the right.
enum {left, right} side;
Node *p = n->getParent();
if (p != nullptr && p->getLeftChild() == n)
side = left;
else
side = right;
// Get the node's right subtree as the new root
// and that subtree's left subtree. Make that
// left subtree the node's new right. Put our
// original node as the left subtree of our
// new root.
Node *temp = n->getRightChild();
n->setRightChild(temp->getLeftChild());
temp->setLeftChild(n);
// Fix the original parent to point to the new
// root. If there isn't a parent, update the
// actual tree root. Otherwise, there is a
// parent so if we were the left subtree, update
// it, otherwise, the right.
if (p == nullptr)
setRoot(temp);
else if (side == left)
p->setLeftChild(temp);
else
p->setRightChild(temp);
} // rotateLeft
// --------------------------------------------------
// Rotate the subtree left.
void AVLTree::rotateRight(Node *n) {
// Get the node's parent and if it exists and the
// node was it's left subtree, remember we are
// processing the left, otherwise, the right.
enum {left, right} side;
Node *p = n->getParent();
if (p != nullptr && p->getLeftChild() == n)
side = left;
else
side = right;
// Get the node's left subtree as the new root
// and that subtree's right subtree. Make that
// right subtree the node's new left. Put our
// original node as the right subtree of our
// new root.
Node *temp = n->getLeftChild();
n->setLeftChild(temp->getRightChild());
temp->setRightChild(n);
// Fix the original parent to point to the new
// root. If there isn't a parent, update the
// actual tree root. Otherwise, there is a
// parent so if we were the left subtree, update
// it, otherwise, the right.
if (p == nullptr)
setRoot(temp);
else if (side == left)
p->setLeftChild(temp);
else
p->setRightChild(temp);
} // rotateRight
// --------------------------------------------------
// Set the root. Change the tree root to the node
// and if it exists, remove it's parent.
void AVLTree::setRoot(Node *n) {
root = n;
if (root != nullptr)
root->removeParent();
} // setRoot
// --------------------------------------------------
// Figure out the default spacing for element. Each
// higher level doubles the preceeding. The bottom
// level is one.
int AVLTree::spacing(int level) {
int result = 2;
for (int i = 0; i < level; i++)
result += result;
return result;
} // spacing
// **************************************************
// Test the class.
int main() {
// Allocate an array to keep track of the data we
// add to the tree, initialize the random numbers,
// allocate an empty tree.
int data[10];
srand(time(0));
AVLTree *tree = new AVLTree();
// Insert 10 unique random numbers into the tree.
// For each number we are adding, attempt to insert
// a random number, until it works because it is
// unique. Afterwards, display the new number and
// the current state of the tree.
for (int i = 0; i < 10; i++) {
while (!tree->insert(data[i] = rand()%100));
std::cout << "Adding " << data[i] << std::endl;
tree->print();
} // for
// Remove each of the numbers by displaying the
// number being removed, performing the removal,
// and displaying the current state of the tree.
for (int i = 0; i < 10; i++) {
std::cout << "Removing " << data[i] << std::endl;
tree->remove(data[i]);
tree->print();
} // for
return 0;
} // main
]]>
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