29.1.5. Question 4

A tree is a connected acyclic graph, that means it is a graph which is connected, that means you take every pair of distinct nodes, there will be a path and it is acyclic, that means the graph has no cycle.

A tree is defined as a type of graph that is both connected and acyclic. To understand this, let's break down the terms: 'connected' means that every pair of nodes (or vertices) in the tree can be reached from one another by traversing edges. 'Acyclic' means that there are no cycles in the graph, which are paths that start and end at the same node and include at least three edges. In simpler terms, a tree looks like a branching structure without any loops.

Think of a tree as a family tree, where each person is a node and the lines connecting them are relationships (edges). Just like you can trace a path between any two family members, you can follow connections in a tree. However, there are no loops - for example, you can't go from your grandmother to your father and then back to your grandmother without repetition.

We have to show that if you take any tree with n nodes, then the tree has n - 1 edges.

To prove that a tree with n nodes has exactly n - 1 edges, we use mathematical induction. First, we establish a base case: a tree with 1 node has 0 edges, which fits the n - 1 formula. Next, we assume it’s true for all trees with up to k nodes (the inductive hypothesis). When we add one more node to create a tree with k + 1 nodes, we can always connect this new node to the existing tree with one edge. This new edge does not create a cycle, maintaining the tree's properties. Therefore, if the previous tree had k - 1 edges, adding one more gives us k edges, which maintains our formula.

Consider a city where nodes represent locations and edges represent roads connecting them. If you have 5 unique locations (nodes) in this city, you can connect all of them with 4 roads (edges) without any loops. Adding another location would mean you need to add one more road to connect it without creating any circular routes, leading us back to our formula that for 5 locations, there are 4 roads.

Assume the inductive hypothesis is true, that means assume the statement is true for all trees consisting of up to k nodes. And now we are going to the inductive step, where we are going to consider an arbitrary tree consisting of k + 1 nodes.

In the inductive step of our proof, we take the assumption that trees with up to k nodes follow the rule of having n - 1 edges. Now, we need to consider a tree with k + 1 nodes. By removing an edge from the tree that connects two nodes, we divide the tree into two smaller trees, each of which must also adhere to the edge rule because they are trees with fewer than k + 1 nodes. Therefore, each of these trees has their corresponding nodes minus one edges.

Imagine you have a small orchard of k trees (nodes), and you have connected them in such a way (edges) that there are no paths looping back on themselves. When you add another tree, you dig one path to connect this new tree to one of the existing trees without creating any loops. Each time you do this, you may visualize shrinking the orchard after taking away the path and you are left with a smaller version of your initial setup where this rule holds true.

My claim is that the edge (u, v) is a cut edge in the tree and this is true for any edge in the tree.

A cut edge (or bridge) is an edge in a graph whose removal increases the number of connected components. In a tree, every edge is a cut edge because removing any edge will disconnect the tree into two separate parts. This is significant because it reinforces the idea that trees are minimally connected; they can remove an edge and still remain connected via other nodes, and each edge is essential for maintaining that connection.

Imagine a string of fairy lights where each light (node) is connected through wires (edges). If you cut a wire (remove an edge), the string splits into two parts, effectively removing the link between all the lights that were once connected. In this case, each wire is essential, demonstrating the tree's structure where removing any edge will cut the connection.

So, as per the inductive hypothesis, I can apply the inductive hypothesis and claim that the number of edges in C is n - 1, the number of edges in C is n - 1.

By applying the inductive hypothesis, we know that the two smaller trees (C1 and C2) each contain n - 1 edges. Thus, the total number of edges in the larger tree G can be expressed mathematically as: total edges = (n1 - 1) + (n2 - 1) + 1 (the edge connecting the two). This simplifies to n1 + n2 - 1, which equals (k + 1 - 1) or k edges in total. This confirms our original hypothesis about the relationship between nodes and edges in a tree.

Returning to our fairy lights analogy, if you split the lights into two smaller segments before re-joining them together, each segment functions independently without carrying any loops. Each segment follows its connection rule of n - 1 lights for each segment, hence maintaining the principle that together they still adhere to the overall structure of the string of lights.

--