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## 'Tree Graphs' printed from http://nrich.maths.org/

The simplest tree graph consists of one line with two vertices,
one at each end.

If a new line is added it must connect to one and only one of
the existing vertices.

If the new line connected to no vertices, the tree graph would
not be connected, as the new line's vertices could not be reached
from the existing vertices.

If each end of the new line connects to a vertex the graph will
have a circuit and will not be a tree graph.

So every new line added will join on to one existing vertex and
create a new vertex at its end. This adds one line and one vertex
to the tree graph making no change to the difference between the
numbers of edges and vertices. So the difference remains constant
at what it originally was.

Marcos's solution is a subtle variation
on the above method:

- Firstly, it's obvious that every edge is connected to exactly
two vertices. (by definition)
- More importantly, there is at least one vertex which is
connected to exactly one edge.

**Proof** . If there wasn't at least one such
vertex we could keep moving around the graph indefinitely and as
there is a finite number of edges it would mean that there is a
cycle, counter to the definition of a tree.

Take one such vertex as described in (2) and its respective
edge. So far we have 2 vertices and 1 edge. The difference is
1.

(*) Add to this an adjacent edge (this is adding one edge and
one vertex. The difference is still one.

Generally, carrying out this step, (*), an arbitrary number of
times until we add the final edge will still result in a difference
of 1 (as the step was in no way linked to fact that the previous
edge was the starting one)

Hence, the number of vertices is one more than the number of
edges.