37. How many rooted and unrooted phylogenetic trees, respectively, are possible with four different sequences?
(A) 3 and 15
(B) 15 and 3
(C) 15 and 12
(D) 12 and 3
Answer: (A) 3 and 15
For four distinct sequences (labeled taxa A, B, C, D), there are 15 possible rooted binary phylogenetic trees and 3 possible unrooted binary phylogenetic trees.
Rooted Trees Calculation
Rooted trees specify a directional evolutionary history from a common ancestor. The formula for the number of rooted binary trees with n labeled taxa is the double factorial (2n-3)!!, equivalent to the product of all odd numbers up to (2n-3): (2n-3) × (2n-5) × … × 3 × 1.
(2×4-3)!! = 5!! = 5 × 3 × 1 = 15
Each unrooted tree can have the root placed on any of its (2n-3) = 5 internal branches, confirming 3 unrooted trees × 5 = 15 rooted trees.
Unrooted Trees Calculation
Unrooted trees represent branching patterns without a specified root. For n labeled taxa, the number is (2n-5)!!.
(2×4-5)!! = 3!! = 3 × 1 = 3
The three topologies are: ((A,B),(C,D)), ((A,C),(B,D)), and ((A,D),(B,C)).
Option Analysis
| Option | Unrooted | Rooted | Status |
|---|---|---|---|
| (A) | 3 | 15 | ✅ Correct – matches standard formulas |
| (B) | 15 | 3 | ❌ Incorrect – reverses the counts |
| (C) | 15 | 12 | ❌ Incorrect – 12 lacks basis in formulas |
| (D) | 12 | 3 | ❌ Incorrect – 12 overstates unrooted trees |
SEO-Friendly Article: Rooted and Unrooted Phylogenetic Trees with Four Sequences
In bioinformatics and molecular phylogenetics, understanding rooted and unrooted phylogenetic trees for four sequences is essential for reconstructing evolutionary relationships from DNA or protein data. For four distinct taxa (e.g., sequences A, B, C, D), exactly 15 rooted trees and 3 unrooted trees are possible in binary topologies—the standard assumption in tools like MEGA or PHYLIP.
Rooted Phylogenetic Trees Explained
Rooted trees designate an ancestral node, implying directionality. The count follows (2n-3)!!: for n=4, 5×3×1=15 distinct labeled topologies. This scales rapidly (e.g., 105 for n=5), highlighting computational challenges in tree search algorithms.
Unrooted Phylogenetic Trees Formula
Unrooted trees omit the root, focusing on splits. Use (2n-5)!!: for n=4, 3×1=3 topologies. Each corresponds to one of three quartet resolutions, visualized as a star with three bifurcating patterns.
MCQ Breakdown and Exam Tips
Common in exams like GATE BT or CSIR NET:
- Memorize formulas: Unrooted = (2n-5)!!; Rooted = (2n-3)!!
- Quick check: Rooted trees always exceed unrooted trees for n > 2
- Verification: Rooted = Unrooted × (2n-3)
Final Answer: (A) 3 unrooted, 15 rooted phylogenetic trees
