# What's the difference between a binary search tree and a binary heap?

These two seem very similar and have almost an identical structure. What's the difference? What are the runtime complexities of each?

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With data structure one has to distinguish levels of concern.

One. The abstract data structures (objects stored, their operations) in this question are different. One implements a priority queue, the other a set. A priority queue is not interested in finding an arbitrary element, only the one with largest priority.

Two. The concrete implementation of the structures. Here on first sight both are (binary) trees, with different structural properties though. Both the relative ordering of the keys and the possible global structures differ. (Somewhat imprecise, in a bst keys are ordered left-to-right, in a heap they are ordered top-down.) As IPlant correctly remarks an heap should also be "complete".

Three. There is a final difference in the low level implementation. A (unbalanced) binary search-tree has a standard implementation using pointers. A binary heap to the contrary has an efficient implementation using an array (precisely because of the restricted structure).

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Heap just guarantees that elements on higher levels are greater (for max-heap) or smaller (for min-heap) than elements on lower levels, whereas BST guarantees order (from "left" to "right"). If you want sorted elements, go with BST.by Dante is not a geek

Heap is better at findMin/findMax (O(1)), while BST is good at all finds (O(logN)). Insert is O(logN) for both structures. If you only care about findMin/findMax (e.g. priority-related), go with heap. If you want everything sorted, go with BST.

by xysun

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I think BST is better in findMin & findMax stackoverflow.com/a/27074221/764592 –  Yeo Nov 22 '14 at 5:50

Both binary search trees and binary heaps are both in the set of trees and are tree-based data structures.

Heaps require the nodes to have a priority over their children. In a max heap, each node's children must be less than itself. This is the opposite for a min heap:

Binary search trees (BST) follow a specific ordering (pre-order, in-order, post-order) among sibling nodes. The tree must be sorted, unlike heaps:

BST have average of $O(\log n)$ for insertion, deletion, and search.
Binary Heaps have average $O(1)$ for findMin/findMax and $O(\log n)$ for insertion and deletion.

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Welcome to Computer Science! Note that you can use LaTeX here to typeset mathematics in a more readable way. See here for a short introduction. –  FrankW Jun 20 '14 at 7:17
Insertion into binary heaps is $O(\log n)$, not $O(1)$. What is $O(1)$ is extract_min/extract_max. –  FrankW Jun 20 '14 at 7:18

On top of the previous answers, the heap must have the heap structure property; the tree must be full, and the bottom most layer, which cannot always be full, must be filled leftmost to rightmost with no gaps.

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A Heap a number on top MUST be bigger than it's children.

A BST the numbers to the left MUST be smaller than its parent. The numbers on the right must be bigger than its parent.

A BST is MUCH better for a search where you must find a specific number in general IMO. Since it slowly pinpoints where the number is if it exists.

A Heap on the other hand you will have to check both each side because the number/value you are looking for may exist else where. WHILE on a BST if it isn't in a particular place you know it doesn't exist in this tree.

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What does this add to the previous answers? –  dfeuer Mar 18 at 1:00
Thanks for taking the time to contribute an answer but that time would have been better spent writing something that added to the existing answers, rather than duplicating them. It's also not very useful to inject opinion ("IMO") where concrete performance analysis could have been used (general find in a BST is $O(\log n)$ vs $O(n)$ in a heap) or to describe searching in a BST ($O(\log n)$) as "slow". –  David Richerby Mar 18 at 7:43
This was answered the day before. It's over already. –  bfs Mar 18 at 23:38