Applications of Trees

• Binary search trees
• Recursion trees
• DFS and BFS trees
• Binary heaps
• Huffman Codes
• etc.

Most of these trees had a preferred root, and a preferred direction (away from the root).

Tree: General Definition

A tree is a connected, acyclic graph.

By default, a generic tree is undirected. To understand this definition, we need to be precise about the meaning of acyclic.

• A walk in an undirected graph \(G\) is a sequence of vertices, where each adjacent pair of vertices are adjacent in \(G\).
• A walk is closed if it starts and ends at the same vertex.
• A walk is called a path if it visits each vertex at most once.
• A cycle is a closed walk that enters and leaves each vertex at most once.
• An undirected graph is acyclic if no subgraph is a cycle.

Unravel the definitions

1. Draw an example of a walk that is not a path.

2. Draw an example of a closed walk that is not a cycle.

3. What is a closed path? Can you draw an example?

4. Draw a tree with 6 vertices. How many edges does it have?

5. How many edges does a tree with \(n\) vertices have? Can you prove your answer?

Table Groups

Unique Paths

Lemma. Suppose \(T\) is a tree, and \(u\) and \(v\) are two vertices in \(T\). Then there is a unique path from \(u\) to \(v\).

• There exists a path from \(u\) to \(v\) because \(T\) is connected.
• Suppose to the contrary that there are two different paths from \(u\) to \(v\).
  • Path 1: \(a_0 = u, a_1, a_2, \ldots, a_k = v\).
  • Path 2: \(b_0 = u, b_1, b_2, \ldots, b_j = v\).
    • Let \(a_s = b_s\) be the last vertex at the start of these paths that are the same.
    • Let \(a_{k-e} = b_{j-e}\) be the first vertex at the end of these paths that are the same.
  • Then \(a_s, \ldots, a_{k-e}, b_{j-e-1}, \ldots, b_s\) is a cycle, contradicting that \(T\) is a tree.

Equivalent Condition

The converse of the Lemma is also true: If \(G\) is a graph with the property that every pair of vertices are joined by a unique path, then \(G\) is a tree.

• Connected: \(\checkmark\)
• Acyclic: A cycle would contradict the unique path condition.

Weighted Edges

A graph is called weighted if there is an assignment of real numbers (weights) to the edges. For example,

• Cost function in a network.
• Distance function in a spatial arrangement.

Minimum Spanning Tree of a Weighted Graph

Given a weighted graph \(G\), a minimum spanning tree \(T\) is a subgraph of \(G\) such that:

• \(T\) contains all the vertices of \(G\), and
• The sum of weights of the edges of \(T\) is as small as possible.

6. Why do the above two conditions guarantee that \(T\) is a tree?

7. Draw an example of a weighted graph with two different minimum spanning trees (if possible).

8. Draw an example of a weighted graph with a unique smallest edge \(e\), such that the minimum spanning tree does not contain \(e\) (if possible).

Minimum spanning is unique (almost)

Lemma. If all edge weights in a connected graph \(G\) are distinct, then \(G\) has a unique minimum spanning tree.

• Suppose \(G\) has two minimum spanning trees \(T\) and \(T'\).
• Let \(e\) be the smallest edge in \((T\setminus T')\cup(T'\setminus T)\).
  • WLOG, \(e\in T\setminus T'\).
• Adding this edge to \(T'\) creates a cycle.
  • One of the edges in this cycle is not in \(T\).
  • Exchange this edge for \(e\), and you reduce the weight of \(T'\), a contradiction.