This document discusses different types of trees and graphs. It defines properties of trees including that trees are connected graphs with n-1 edges and no cycles. It describes binary search trees where all descendants to the left of a node are smaller and all to the right are greater. The document also covers heaps, tree traversals, common tree algorithms and their strategies, and briefly mentions graphs.

1. Trees and Graphs
Spencer Moran
12/3/13

2. Properties of a Tree
• Tree is a graph where:
– n-1 edges
– Connected
– No cycles
• If two of the above are true, then all 3 are
true.
• Tree questions are usually asked to test your
understanding of recursion.

3. Binary Search Trees
• Binary tree is one where each node as at most
2 children.
• Binary Search tree is a binary tree where all
the descendants to the left of a node are
smaller than it and all the descendants to the
right are greater than it.
• Most interview questions focus on binary or
binary search trees.

4. Binary Search Tree Complexity
•
•
•
•
•
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Lookup: O(log n)
Insertion: O(log n)
Deletion: O(log n)
Smallest/Largest element: O(n)
Traversal: O(n)
Print in sorted order: O(n)

5. Heaps
• (Usually) binary tree where each node’s
children are smaller (or larger in a Min-heap)
than it.
• Allows for constant time extraction of the
largest element in the tree.
• Lookup becomes O(n)
• Used if extracting the smallest or largest node
is paramount

6. Searching
BFS
• Queue
• Searches minimum
depths first, always
searches lowest depth
last
• High memory
requirement
• Shortest path
DFS
• Stack
• Explores full depth,
doesn’t search any
particular depth last
• Low memory
requirement

7. Traversals
• Pre-order:
– Explore node, then left children, then right
• In-order:
– Explore left children, then node, then right
• Post-order:
– Explore left children, then right, then node
• Implemented using simple recursion

8. Preorder Traversal

9. Sample Question
Find the height of a binary tree.

10. Strategy
• Use recursion
• Think of each child as its own separate subtree
with that node as the root
• Return 1 + the max of left subtree’s height and
right subtree’s height

12. Sample Question
Given the value of two nodes in a BST, find the
lowest common ancestor. You may assume both
values exist in the tree.

13. Strategy
• Uses the left/right descendant property of a
BST to traverse through the nodes.
• Starting at the root, traverse through the tree.
• If both values are greater than the current
node, go right.
• If both are smaller than the current, go left.
• LCA will be the first node you encounter that
is between the two nodes you are given.

15. Other Common Tree Questions
• Binary tree to heap
• Is Binary tree a BST?
• Balance an unbalanced BST (left or right
rotation)
• Count # of leaves

16. Graphs
• Graph questions are much more complicated
and less common.
• We will cover graphs during the group
meeting.