Question:
If there are three unrooted trees for four protein sequences, the number of rooted trees for the same number of sequences is ________________.
If there are three unrooted trees for four protein sequences, the number of rooted trees for the same number of sequences is ________________.
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Remember: rooted trees \((2n-3)!!\); unrooted trees \((2n-5)!!\) for \(n\) labeled taxa.
Updated On: Aug 26, 2025
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Correct Answer: 15
Solution and Explanation
Step 1: The number of unrooted binary trees for \(n\) taxa is \((2n-5)!!\). For \(n=4\), \((2\cdot4-5)!!=3!!=3\), matching the given.
Step 2: The number of rooted binary trees for \(n\) taxa is \((2n-3)!!\). Thus, for \(n=4\): \[ (2\cdot4-3)!!=5!!=5\cdot3\cdot1=\boxed{15}. \]
Step 2: The number of rooted binary trees for \(n\) taxa is \((2n-3)!!\). Thus, for \(n=4\): \[ (2\cdot4-3)!!=5!!=5\cdot3\cdot1=\boxed{15}. \]
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