Q.51 If there are three unrooted trees for four protein sequences, the number of rooted
trees for the same number of sequences is ___________.
Phylogenetic Trees for Four Protein Sequences
For four labeled taxa (protein sequences), there are exactly
3 unrooted binary phylogenetic trees and
15 rooted binary phylogenetic trees.
Tree Counting Formulas
Phylogenetic trees describe evolutionary branching among sequences.
- Unrooted trees show branching without directionality
- Rooted trees specify an ancestor (root)
The formulas for labeled binary (fully resolved) trees are:
Unrooted Trees
(2n − 5)!!
For n = 4:
(2×4 − 5)!! = 3!! = 3 × 1 = 3
Rooted Trees
(2n − 3)!!
For n = 4:
(2×4 − 3)!! = 5!! = 5 × 3 × 1 = 15
Why 15 Rooted Trees?
Each unrooted tree has 5 edges. Placing a root on any edge gives:
3 unrooted × 5 edges = 15 rooted
Visualizing Unrooted Tree Types
- Central 4-way topology
((A,B),(C,D))split(A,(B,(C,D)))and symmetry variants
Rooted trees add direction (ancestral root) and generate 15 distinct topologies.
Correct Answer
Number of rooted binary trees for four taxa = 15
Common Confusions
- 3 — unrooted only
- 12 — miscounts unlabeled trees
- 105 — number for 5 taxa
Introduction: Unrooted Trees for Four Protein Sequences
In biotechnology and computational biology, phylogenetic trees are essential tools for modeling
evolutionary relationships. A common exam question asks how many possible trees can be constructed
from four labeled protein sequences. While only 3 unrooted binary trees exist, these expand into
15 possible rooted trees.
Core Formulas Explained
For n = 4:
Unrooted trees
(2n − 5)!! = (2×4 − 5)!! = 3!! = 3
Rooted trees
(2n − 3)!! = (2×4 − 3)!! = 5!! = 15
Each unrooted tree yields five rooted topologies by selecting each edge as a root point.
Exam Relevance
Biotechnology entrance tests like GATE BT and IIT JAM frequently ask:
“If 3 unrooted trees exist for four sequences, how many rooted trees are possible?”
The correct answer is 15.
Final Result: 15 rooted phylogenetic trees
