Exact formulas for the variance of several balance indices under the Yule model

(By G. Cardona, A. Mir, F. Rosselló)

Supplementary Material

• Computations for all trees with number of leaves up to 9:
  • Python script used to compute all pairs of trees and find the exact value for the variances and covariances in the paper. It also gives, inside parenthesis, the value predicted by the formulas in the paper. It uses the package phylonetwork.
  • Results given by the script:

    3
    E(sackin) = 5.000000 (5.000000)
    E(cophenetic) = 1.000000 (1.000000)
    E(cophebar) = 6.000000 (6.000000)
    E(colless) = 1.000000 (1.000000)
    V(sackin) = 0.000000 (0.000000)
    V(cophenetic) = 0.000000 (0.000000)
    V(cophebar) = 0.000000 (0.000000)
    V(colless) = 0.000000 (0.000000)
    Cov(sackin,cophenetic) = 0.000000 (0.000000)
    Cov(sackin,cophebar) = 0.000000 (0.000000)
    4
    E(sackin) = 8.666667 (8.666667)
    E(cophenetic) = 3.333333 (3.333333)
    E(cophebar) = 12.000000 (12.000000)
    E(colless) = 2.000000 (2.000000)
    V(sackin) = 0.222222 (0.222222)
    V(cophenetic) = 0.888889 (0.888889)
    V(cophebar) = 2.000000 (2.000000)
    V(colless) = 2.000000 (2.000000)
    Cov(sackin,cophenetic) = 0.444444 (0.444444)
    Cov(sackin,cophebar) = 0.666667 (0.666667)
    5
    E(sackin) = 12.833333 (12.833333)
    E(cophenetic) = 7.166667 (7.166667)
    E(cophebar) = 20.000000 (20.000000)
    E(colless) = 3.500000 (3.500000)
    V(sackin) = 0.805556 (0.805556)
    V(cophenetic) = 5.138889 (5.138889)
    V(cophebar) = 10.000000 (10.000000)
    V(colless) = 3.250000 (3.250000)
    Cov(sackin,cophenetic) = 2.027778 (2.027778)
    Cov(sackin,cophebar) = 2.833333 (2.833333)
    6
    E(sackin) = 17.400000 (17.400000)
    E(cophenetic) = 12.600000 (12.600000)
    E(cophebar) = 30.000000 (30.000000)
    E(colless) = 5.000000 (5.000000)
    V(sackin) = 1.840000 (1.840000)
    V(cophenetic) = 17.040000 (17.040000)
    V(cophebar) = 30.000000 (30.000000)
    V(colless) = 6.800000 (6.800000)
    Cov(sackin,cophenetic) = 5.560000 (5.560000)
    Cov(sackin,cophebar) = 7.400000 (7.400000)
    7
    E(sackin) = 22.300000 (22.300000)
    E(cophenetic) = 19.700000 (19.700000)
    E(cophebar) = 42.000000 (42.000000)
    E(colless) = 6.833333 (6.833333)
    V(sackin) = 3.387778 (3.387778)
    V(cophenetic) = 42.787778 (42.787778)
    V(cophebar) = 70.000000 (70.000000)
    V(colless) = 10.072222 (10.072222)
    Cov(sackin,cophenetic) = 11.912222 (11.912222)
    Cov(sackin,cophebar) = 15.300000 (15.300000)
    8
    E(sackin) = 27.485714 (27.485714)
    E(cophenetic) = 28.514286 (28.514286)
    E(cophebar) = 56.000000 (56.000000)
    E(colless) = 8.666667 (8.666667)
    V(sackin) = 5.494240 (5.494240)
    V(cophenetic) = 90.522812 (90.522812)
    V(cophebar) = 140.000000 (140.000000)
    V(colless) = 15.765079 (15.765079)
    Cov(sackin,cophenetic) = 21.991474 (21.991474)
    Cov(sackin,cophebar) = 27.485714 (27.485714)
    9
    E(sackin) = 32.921429 (32.921429)
    E(cophenetic) = 39.078571 (39.078571)
    E(cophebar) = 72.000000 (72.000000)
    E(colless) = 10.750000 (10.750000)
    V(sackin) = 8.193827 (8.193827)
    V(cophenetic) = 170.350969 (170.350969)
    V(cophebar) = 252.000000 (252.000000)
    V(colless) = 21.089881 (21.089881)
    Cov(sackin,cophenetic) = 36.727602 (36.727602)
    Cov(sackin,cophebar) = 44.921429 (44.921429)