# Lecture 8: Sorting Trees

We implemented mergesort for trees.
We showed that there is considerable opportunity for parallelism compared with mergesort for lists.
We mentioned the need for tree rebalancing, in order to constrain the depth of trees returned by recursive class in our mergesort.

### Key Concepts

• Sorted Trees
• Divide and Conquer for Trees
• Span Analysis
• Tree Balance

A tree is said to be balanced  if d is O(log n), with d the depth of the tree and with n the size of the tree. The size could be measured either in terms of the number of nodes or the number of leaves (e.g., Emptys), depending on the particular application code. (A binary tree has n nodes if and only if it has n+1 leaves.) For a balanced tree we often view the left subtree and right subtrees of the root as each containing roughly half the overall number of nodes, i.e., roughly n/2. We say that a tree of depth d is perfectly balanced if each immediate subtree is perfectly balanced and has depth d-1. We say that a tree is unbalanced if it is not balanced.