To determine the total number of vertices in the tree, we sum the number of all vertices in the tree.

Explanation:

• Root: The single topmost vertex of the tree.
  $\text{Count} = 1$
• Parents: Nodes that have at least one child.
  $\text{Count} = 6$
• Children: Nodes that are direct descendants of parents.
  $\text{Count} = 12$
• Leaves: Nodes that do not have any children (endpoints of the tree).
  $\text{Count} = 7$

However, we must avoid double-counting:

1. The root is also a parent, so it’s already included in the parent count.
2. Children may or may not be leaves.

The total number of vertices is the union of all the unique nodes across the categories. We calculate this as follows:

Formula:

$\text{Total Vertices} = \text{Root} + (\text{Parents (excluding root)}) + \text{Children} + \text{Leaves (excluding already counted nodes})$

In this question, summing all provided unique categories directly suffices because all are distinct:

$\text{Total Vertices} = 1 + 6 + 12 + 7 = 26$

Final Answer:

26

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Would you like me to explain the reasoning behind trees in graph theory or clarify anything further?

Related Questions:

1. How do parent-child relationships determine tree structure?
2. What is the difference between internal nodes and leaves in a tree?
3. Why is the root not counted as a child or leaf?
4. Can a tree have nodes that are neither parents nor leaves?
5. How does a binary tree compare to this structure?

Tip:

In any tree, the number of edges is always one less than the total number of vertices:
$\text{Edges} = \text{Vertices} - 1$