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cs2381 Notes: 20 Binary Trees

··2 mins

Binary Trees

We’ve looked at singly linked lists (ConsList). Each cell in a ConsList has a reference to one other cell. This produces a sequence of nodes.

What if we have nodes with more than one reference to other nodes?

The general case of this is a structure called a graph.

(Draw some graphs.)

Arbitrary graphs can be useful, but they take some effort to deal with. Minimally, graphs can have cycles, which means that just trying to do a simple traverasal can lead to an infinite loop.

A ConsList is the simplest example of a class of useful structures: trees.

Trees are:

  • graphs
  • with a single root
  • with no cycles

A simple, commonly used tree is the Binary Tree, where each node has (up to) two references to other nodes.

Binary Trees #

Without getting more specific, there are some things we can say about binary trees:

  • There’s one root.
  • Each node has two references to other nodes, those nodes are its “children”.
  • That means every non-root node has one parent.
  • Nodes with children are branches.
  • Nodes without children are leaves.
  • The height of a tree is the maximum length of a path from the root to a leaf.

The maximum height of a tree with n items is n.

The minimum height of a tree with n items is log(n).

What’s a log?

  • Inverse of exponent
  • specifically, we’re talking about log base 2, log_2(n).
  • How many digits do we need to represent the number (base 2, so binary digits).
  • This grows very slowly. O(log(n)) is much more efficient than O(n).

One kind of minimum height tree is called a complete tree:

  • All of the levels are full.
  • Except maybe the last level, which is only missing items on the right.

In CS, trees grow down.

At each level, numbering from zero, there’s space for 2^x items.

So a full tree of height h has 2^h items on the bottom level, 2^(h-1) on the previous level, etc for a total of 2^(h+1) - 1 items.

Tree Traversals #

Commonly, we store values in trees by having each node have an associated value.

That gives us some questions on how to traverse the tree and visit the values.

  • Pre-order: Visit node before children.
  • Post-order: Visit node after children.
  • In-order: Visit node between left and right trees.