How to prove all trees have at least as many leaves as its maximum degree?

Question

EDIT:

I think I found the issue. $m_h = n - (\Delta T + 1)$, true. But the order of $H$ is not $n$, it's $n_H = n-1$. So $m_H = n_h - \Delta T$. Therefore, replacing all instances of $\Delta T + 1 below this point with $\Delta T$ will result in the wanted solution.

Original post:

I want to show this:

Prove that if $T$ is a tree, then $T$ has at least $\Delta(T)$ leaves.

I read this similar question, but ended up with showing $T$ has at least $\Delta(T)+1$ leaves. While this technically proves the original statement, that leads me to wonder why then that the statement does not say:

Prove that if $T$ is a tree, then $T$ has at least $\Delta(T)+1$ leaves instead.

My attempt to prove the original proposition (number of leaves $\geq\Delta(T)$).

Proof. Let $T$ be an order $n$ tree with maximum degree $\Delta(T)$. The size of $T$ must be $m=n-1$ by definition of a tree. We want to show $T$ must have at least $\Delta(T)$ leaves.

Set $v\in T$ be some vertex with $\deg(v)=\Delta(T)$. Define $H = T-v$ be an induced subgraph of $T$ with $v$ removed. Then the size of $H$ must be \begin{align} m_H &= m - \Delta(T)\\ &= n - 1 - \Delta(T)\\ &= n - (\Delta(T) + 1) \end{align}

So $H$ is a forest of order $n_H = n-1$ with size $m_H = n - (\Delta T + 1)$. By definition of a forest, $H$ has $k = \Delta(T) + 1$ components.

Let $k_1$ be some component of $H$. If $k_1$ has only one vertex $v_1$, then $v_1$ was a neighbor of $v$ and a leaf of $T$ (because $\deg_H(v_1)=0\implies \deg(v_1) = 0 + 1 = 1$ before removing $v$). Let $i$ denote the number of such components.

Choose $k_2$ from the remaining $k-i$ components. $k_2$ must have 2 or more vertices (else it would follow under $k_1$'s rules). Because $k_2$ is connected and acyclic (as $T$ was an acyclic tree), $k_2$ is a tree. Therefore, $k_2$ has at least 2 leaves, only one of which could be a neighbor of $v$. Therefore, $k_2$ contains at least one leaf of $T$.

Thus, the components of $H$ each contribute at least one leaf to $T$.

So $T$ has at least $k = \Delta T + 1$ leaves.

$\square$

Where have I gone wrong?

Answer 1

See my edit above.

Proof. Let $T$ be an order $n$ tree with maximum degree $\Delta(T)$. The size of $T$ must be $m=n-1$ by definition of a tree. We want to show $T$ must have at least $\Delta(T)$ leaves.

Set $v\in T$ be some vertex with $\deg(v)=\Delta(T)$. Define $H = T-v$ be an induced subgraph of $T$ with $v$ removed. Then the size of $H$ must be \begin{align} m_H &= m - \Delta(T)\\ &= n - 1 - \Delta(T)\\ &= n - (\Delta(T) + 1) \end{align}

So $H$ is a forest of order $n_H = n-1$ with size $m_H = n - (\Delta T + 1) = n_H - \Delta T$. By definition of a forest, $H$ has $k = \Delta(T)$ components.

Let $k_1$ be some component of $H$. If $k_1$ has only one vertex $v_1$, then $v_1$ was a neighbor of $v$ and a leaf of $T$ (because $\deg_H(v_1)=0\implies \deg(v_1) = 0 + 1 = 1$ before removing $v$). Let $i$ denote the number of such components.

Choose $k_2$ from the remaining $k-i$ components. $k_2$ must have 2 or more vertices (else it would follow under $k_1$'s rules). Because $k_2$ is connected and acyclic (as $T$ was an acyclic tree), $k_2$ is a tree. Therefore, $k_2$ has at least 2 leaves, only one of which could be a neighbor of $v$. Therefore, $k_2$ contains at least one leaf of $T$.

Thus, the components of $H$ each contribute at least one leaf to $T$.

So $T$ has at least $k = \Delta T$ leaves.

$\square$