How many edges will a tree consisting of N nodes have?ans.NN + 1Log(N)N – 1 Previous Marked for Review Next

To determine the number of edges in a tree consisting of $N$ nodes, we can apply a fundamental property of trees in graph theory.

Breakdown of the Problem

1. A tree is a connected graph that does not contain cycles.
2. The defining characteristic of a tree is that it has exactly one less edge than the number of nodes.

Relevant Concepts

1. For a tree with $N$ nodes, the number of edges $E$ can be formulated as: $E = N - 1$

Analysis and Detail

1. Let's consider the properties of trees:
   • A tree with 1 node has 0 edges.
   • A tree with 2 nodes has 1 edge.
   • A tree with 3 nodes has 2 edges.
   • Continuing this pattern, we can see that each time we add a node, we add exactly one edge to maintain the single connected component and avoid cycles.

Verify and Summarize

• For any integer $N$, the relationship $E = N - 1$ holds true as it encapsulates the essence of trees being acyclic and connected.

Final Answer

The number of edges in a tree consisting of $N$ nodes is given by $E = N - 1$.