Binary tree

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In computer science, a binary tree is a type of tree (data structure) where each item within the tree has at most two children.

Types of binary trees[change | change source]

A complete binary tree which is not full.
  • In a balanced binary tree the left and right branches of every item differ in height by no more than 1.
  • In a complete binary tree every level, except possibly the last, is completely filled, and all items in the last level are as far left as possible.
  • In a full binary tree every item has either 0 or 2 children.
  • In a perfect binary tree all interior items have two children and all leaves have the same depth or same level. A perfect binary tree is also a full and complete binary tree.

Representations[change | change source]

Array[change | change source]

A binary tree can be implemented using an array by storing its level-order traversal.[1] In a zero-indexed array, the root is often stored at index 1.

For the nth item of the array its:

  • left child is stored at the 2n index.
  • right child is stored at the 2n+1 index.
  • parent is stored at the n/2 index.

References[change | change source]

In a programming language with references, binary trees are typically constructed by having a tree structure which contains references to its left child and its right child.

Traversals[change | change source]

Traversals of an example tree:
Pre-order (red): F, B, A, D, C, E, G, I, H
In-order (yellow): A, B, C, D, E, F, G, H, I
Post-order (green): A, C, E, D, B, H, I, G, F

Pre-order[change | change source]

The current item is visited, then the left branch is visited, and then the right branch is visited.

void preOrder(Item item) {
    if (item == null) return;
    visit(item);
    preOrder(item.left);
    preOrder(item.right);
}

In-order[change | change source]

The left branch is visited, then the current item is visited, and then the right branch is visited.

void inOrder(Item item) {
    if (item == null) return;
    inOrder(item.left);
    visit(item);
    inOrder(item.right);
}

Post-order[change | change source]

The left branch is visited, the right branch is visited, and then the current item is visited.

void postOrder(Item item) {
    if (item == null) return;
    postOrder(item.left);
    postOrder(item.right);
    visit(item);
}

References[change | change source]

  1. Adamchik, Victor. "Binary Heap". Computer Science - 121 Fall 2009. CMU. Retrieved 11 October 2020.