Table 1 Probabilities of 15 rooted gene trees given the phylogenetic network ψ of Fig. 2 b (w=0). The quantity g _ij (t) is the probability that i lineages coalesce into j lineages within time t [36]

T ₁=(((b,c),a),d)

\(g_{21}(y)[\gamma (g_{21}(x)+g_{22}(x)\frac {1}{3})+(1-\gamma)(g_{22}(x)\frac {1}{3})]\)

\(+g_{22}(y)[\gamma ^{2}(g_{31}(x)\frac {1}{3}+g_{32}(x)\frac {1}{3}\frac {1}{3}+g_{33}(x)\frac {1}{6}\frac {1}{3})\)

\(+(1-\gamma)^{2}(g_{32}(x)\frac {1}{3}\frac {1}{3}+g_{33}(x)\frac {1}{6}\frac {1}{3})\)

\(+2\gamma (1-\gamma)(g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3})]\)

T ₂=(((b,c),d),a)

\(g_{21}(y)[(1-\gamma)(g_{21}(x)+g_{22}(x)\frac {1}{3})+\gamma (g_{22}(x)\frac {1}{3})]\)

\(+g_{22}(y)[(1-\gamma)^{2}(g_{31}(x)\frac {1}{3}+g_{32}(x)\frac {1}{3}\frac {1}{3}+g_{33}(x)\frac {1}{6}\frac {1}{3})\)

\(+\gamma ^{2}(g_{32}(x)\frac {1}{3}\frac {1}{3}+g_{33}(x)\frac {1}{6}\frac {1}{3})\)

\(+2\gamma (1-\gamma)(g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3})]\)

T ₃=((a,b),(c,d))

\(g_{22}(y)[(\gamma ^{2}+(1-\gamma)^{2})(g_{32}(x)\frac {1}{3}\frac {1}{3}+g_{33}(x)\frac {2}{6}\frac {1}{3})\)

\(+\gamma (1-\gamma)(g_{21}(x)g_{21}(x)+g_{21}(x)g_{22}(x)\frac {1}{3}+g_{22}(x)g_{21}(x)\frac {1}{3}+g_{22}(x)g_{22}(x)\frac {2}{6}\frac {1}{3})\)

\(+\gamma (1-\gamma)(g_{22}(x)g_{22}(x)\frac {2}{6}\frac {1}{3})]\)

T ₄=((a,c),(b,d))

\(g_{22}(y)[(\gamma ^{2}+(1-\gamma)^{2})(g_{32}(x)\frac {1}{3}\frac {1}{3}+g_{33}(x)\frac {2}{6}\frac {1}{3})\)

\(+\gamma (1-\gamma)(g_{21}(x)g_{21}(x)+g_{21}(x)g_{22}(x)\frac {1}{3}+g_{22}(x)g_{21}(x)\frac {1}{3}+g_{22}(x)g_{22}(x)\frac {2}{6}\frac {1}{3})\)

\(+\gamma (1-\gamma)(g_{22}(x)g_{22}(x)\frac {2}{6}\frac {1}{3})]\)

T ₅=(((a,b),c),d)

\(g_{22}(y)[\gamma ^{2}(g_{31}(x)\frac {1}{3}+g_{32}(x)\frac {1}{3}\frac {1}{3}+g_{33}(x)\frac {1}{6}\frac {1}{3})+(1-\gamma)^{2}(g_{33}(x)\frac {1}{6}\frac {1}{3})\)

\(+\gamma (1-\gamma)(g_{21}(x)g_{22}(x)\frac {1}{3}+g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3})+\gamma (1-\gamma)g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3}]\)

T ₆=(((a,c),b),d)

\(g_{22}(y)[\gamma ^{2}(g_{31}(x)\frac {1}{3}+g_{32}(x)\frac {1}{3}\frac {1}{3}+g_{33}(x)\frac {1}{6}\frac {1}{3})+(1-\gamma)^{2}(g_{33}(x)\frac {1}{6}\frac {1}{3})\)

\(+\gamma (1-\gamma)(g_{21}(x)g_{22}(x)\frac {1}{3}+g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3})+\gamma (1-\gamma)g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3}]\)

T ₇=(a,(b,(c,d)))

\(g_{22}(y)[(1-\gamma)^{2}(g_{31}(x)\frac {1}{3}+g_{32}(x)\frac {1}{3}\frac {1}{3}+g_{33}(x)\frac {1}{6}\frac {1}{3})+\gamma ^{2}(g_{33}(x)\frac {1}{6}\frac {1}{3})\)

\(+\gamma (1-\gamma)(g_{21}(x)g_{22}(x)\frac {1}{3}+g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3})+\gamma (1-\gamma)g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3}]\)

T ₈=(((b,d),c),a)

\(g_{22}(y)[(1-\gamma)^{2}(g_{31}(x)\frac {1}{3}+g_{32}(x)\frac {1}{3}\frac {1}{3}+g_{33}(x)\frac {1}{6}\frac {1}{3})+\gamma ^{2}(g_{33}(x)\frac {1}{6}\frac {1}{3})\)

\(+\gamma (1-\gamma)(g_{21}(x)g_{22}(x)\frac {1}{3}+g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3})+\gamma (1-\gamma)g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3}]\)

T ₉=((a,d),(b,c))

\(g_{21}(y)[\gamma g_{22}(x)\frac {1}{3}+(1-\gamma)g_{22}(x)\frac {1}{3}]\)

\(+g_{22}(y)[\gamma ^{2}(g_{32}(x)\frac {1}{3}\frac {1}{3}+g_{33}(x)\frac {2}{6}\frac {1}{3})+(1-\gamma)^{2}(g_{32}(x)\frac {1}{3}\frac {1}{3}+g_{33}(x)\frac {2}{6}\frac {1}{3})\)

\(+\gamma (1-\gamma)(g_{22}(x)g_{22}(x)\frac {2}{6}\frac {1}{3})+\gamma (1-\gamma)(g_{22}(x)g_{22}(x)\frac {2}{6}\frac {1}{3})]\)

T ₁₀=(((a,b),d),c)

\(g_{22}(y)[\gamma ^{2}(g_{32}(x)\frac {1}{3}\frac {1}{3}+g_{33}(x)\frac {1}{6}\frac {1}{3})+(1-\gamma)^{2}(g_{33}(x)\frac {1}{6}\frac {1}{3})\)

\(+\gamma (1-\gamma)(g_{21}(x)g_{22}(x)\frac {1}{3}+g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3})+\gamma (1-\gamma)(g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3})]\)

T ₁₁=(b,(a,(c,d)))

\(g_{22}(y)[(1-\gamma)^{2}(g_{32}(x)\frac {1}{3}\frac {1}{3}+g_{33}(x)\frac {1}{6}\frac {1}{3})+\gamma ^{2}(g_{33}(x)\frac {1}{6}\frac {1}{3})\)

\(+\gamma (1-\gamma)(g_{21}(x)g_{22}(x)\frac {1}{3}+g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3})+\gamma (1-\gamma)(g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3})]\)

T ₁₂=(((a,d),b),c)

\(g_{22}(y)[\gamma ^{2}(g_{33}(x)\frac {1}{6}\frac {1}{3})+(1-\gamma)^{2}(g_{33}(x)\frac {1}{6}\frac {1}{3})\)

\(+\gamma (1-\gamma)(g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3})+\gamma (1-\gamma)(g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3})]\)

T ₁₃=(((b,d),a),c)

\(g_{22}(y)[\gamma ^{2}(g_{33}(x)\frac {1}{6}\frac {1}{3})+(1-\gamma)^{2}(g_{32}(x)\frac {1}{3}\frac {1}{3}+g_{33}(x)\frac {1}{6}\frac {1}{3})\)

\(+\gamma (1-\gamma)(g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3})+\gamma (1-\gamma)(g_{21}(x)g_{22}(x)\frac {1}{3}+g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3})]\)

T ₁₄=(((a,c),d),b)

\(g_{22}(y)[(1-\gamma)^{2}(g_{33}(x)\frac {1}{6}\frac {1}{3})+\gamma ^{2}(g_{32}(x)\frac {1}{3}\frac {1}{3}+g_{33}(x)\frac {1}{6}\frac {1}{3})\)

\(+\gamma (1-\gamma)(g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3})+\gamma (1-\gamma)(g_{21}(x)g_{22}(x)\frac {1}{3}+g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3})]\)

T ₁₅=(((a,d),c),b)

\(g_{22}(y)[\gamma ^{2}(g_{33}(x)\frac {1}{6}\frac {1}{3})+(1-\gamma)^{2}(g_{33}(x)\frac {1}{6}\frac {1}{3})\)

\(+\gamma (1-\gamma)(g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3})+\gamma (1-\gamma)(g_{22}(x)g_{22}(x)\frac {1}{6}\frac {1}{3})]\)