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]

Gene Tree T i	P(T i|ψ,x,y,γ)
T 1=(((b,c),a),d)	$g_{21}\left(y\right)\left[\gamma \right(g_{21}\left(x\right)+g_{22}\left(x\right)\frac{1}{3})+(1-\gamma \left)\right(g_{22}\left(x\right)\frac{1}{3}\left)\right]$
	$+g_{22}\left(y\right)\left[{\gamma }^2\right(g_{31}\left(x\right)\frac{1}{3}+g_{32}\left(x\right)\frac{1}{3}\frac{1}{3}+g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})$
	$+{(1-\gamma )}^2\left(g_{32}\right(x)\frac{1}{3}\frac{1}{3}+g_{33}(x\left)\frac{1}{6}\frac{1}{3}\right)$
	$+2\gamma (1-\gamma )\left(g_{22}\right(x\left)g_{22}\right(x\left)\frac{1}{6}\frac{1}{3}\right)]$
T 2=(((b,c),d),a)	$g_{21}\left(y\right)\left[\right(1-\gamma \left)\right(g_{21}\left(x\right)+g_{22}\left(x\right)\frac{1}{3})+\gamma (g_{22}\left(x\right)\frac{1}{3}\left)\right]$
	$+g_{22}\left(y\right)\left[{(1-\gamma )}^2\right(g_{31}\left(x\right)\frac{1}{3}+g_{32}\left(x\right)\frac{1}{3}\frac{1}{3}+g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})$
	$+{\gamma }^2\left(g_{32}\right(x)\frac{1}{3}\frac{1}{3}+g_{33}(x\left)\frac{1}{6}\frac{1}{3}\right)$
	$+2\gamma (1-\gamma )\left(g_{22}\right(x\left)g_{22}\right(x\left)\frac{1}{6}\frac{1}{3}\right)]$
T 3=((a,b),(c,d))	$g_{22}\left(y\right)\left[\right({\gamma }^2+{(1-\gamma )}^2\left)\right(g_{32}\left(x\right)\frac{1}{3}\frac{1}{3}+g_{33}\left(x\right)\frac{2}{6}\frac{1}{3})$
	$+\gamma (1-\gamma )\left(g_{21}\right(x\left)g_{21}\right(x)+g_{21}(x\left)g_{22}\right(x)\frac{1}{3}+g_{22}(x\left)g_{21}\right(x)\frac{1}{3}+g_{22}(x\left)g_{22}\right(x\left)\frac{2}{6}\frac{1}{3}\right)$
	$+\gamma (1-\gamma )\left(g_{22}\right(x\left)g_{22}\right(x\left)\frac{2}{6}\frac{1}{3}\right)]$
T 4=((a,c),(b,d))	$g_{22}\left(y\right)\left[\right({\gamma }^2+{(1-\gamma )}^2\left)\right(g_{32}\left(x\right)\frac{1}{3}\frac{1}{3}+g_{33}\left(x\right)\frac{2}{6}\frac{1}{3})$
	$+\gamma (1-\gamma )\left(g_{21}\right(x\left)g_{21}\right(x)+g_{21}(x\left)g_{22}\right(x)\frac{1}{3}+g_{22}(x\left)g_{21}\right(x)\frac{1}{3}+g_{22}(x\left)g_{22}\right(x\left)\frac{2}{6}\frac{1}{3}\right)$
	$+\gamma (1-\gamma )\left(g_{22}\right(x\left)g_{22}\right(x\left)\frac{2}{6}\frac{1}{3}\right)]$
T 5=(((a,b),c),d)	$g_{22}\left(y\right)\left[{\gamma }^2\right(g_{31}\left(x\right)\frac{1}{3}+g_{32}\left(x\right)\frac{1}{3}\frac{1}{3}+g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})+{(1-\gamma )}^2(g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})$
	$+\gamma (1-\gamma )\left(g_{21}\right(x\left)g_{22}\right(x)\frac{1}{3}+g_{22}(x\left)g_{22}\right(x\left)\frac{1}{6}\frac{1}{3}\right)+\gamma (1-\gamma )g_{22}\left(x\right)g_{22}\left(x\right)\frac{1}{6}\frac{1}{3}]$
T 6=(((a,c),b),d)	$g_{22}\left(y\right)\left[{\gamma }^2\right(g_{31}\left(x\right)\frac{1}{3}+g_{32}\left(x\right)\frac{1}{3}\frac{1}{3}+g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})+{(1-\gamma )}^2(g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})$
	$+\gamma (1-\gamma )\left(g_{21}\right(x\left)g_{22}\right(x)\frac{1}{3}+g_{22}(x\left)g_{22}\right(x\left)\frac{1}{6}\frac{1}{3}\right)+\gamma (1-\gamma )g_{22}\left(x\right)g_{22}\left(x\right)\frac{1}{6}\frac{1}{3}]$
T 7=(a,(b,(c,d)))	$g_{22}\left(y\right)\left[{(1-\gamma )}^2\right(g_{31}\left(x\right)\frac{1}{3}+g_{32}\left(x\right)\frac{1}{3}\frac{1}{3}+g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})+{\gamma }^2(g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})$
	$+\gamma (1-\gamma )\left(g_{21}\right(x\left)g_{22}\right(x)\frac{1}{3}+g_{22}(x\left)g_{22}\right(x\left)\frac{1}{6}\frac{1}{3}\right)+\gamma (1-\gamma )g_{22}\left(x\right)g_{22}\left(x\right)\frac{1}{6}\frac{1}{3}]$
T 8=(((b,d),c),a)	$g_{22}\left(y\right)\left[{(1-\gamma )}^2\right(g_{31}\left(x\right)\frac{1}{3}+g_{32}\left(x\right)\frac{1}{3}\frac{1}{3}+g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})+{\gamma }^2(g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})$
	$+\gamma (1-\gamma )\left(g_{21}\right(x\left)g_{22}\right(x)\frac{1}{3}+g_{22}(x\left)g_{22}\right(x\left)\frac{1}{6}\frac{1}{3}\right)+\gamma (1-\gamma )g_{22}\left(x\right)g_{22}\left(x\right)\frac{1}{6}\frac{1}{3}]$
T 9=((a,d),(b,c))	$g_{21}\left(y\right)\left[\gamma g_{22}\right(x)\frac{1}{3}+(1-\gamma \left)g_{22}\right(x\left)\frac{1}{3}\right]$
	$+g_{22}\left(y\right)\left[{\gamma }^2\right(g_{32}\left(x\right)\frac{1}{3}\frac{1}{3}+g_{33}\left(x\right)\frac{2}{6}\frac{1}{3})+{(1-\gamma )}^2(g_{32}\left(x\right)\frac{1}{3}\frac{1}{3}+g_{33}\left(x\right)\frac{2}{6}\frac{1}{3})$
	$+\gamma (1-\gamma )\left(g_{22}\right(x\left)g_{22}\right(x\left)\frac{2}{6}\frac{1}{3}\right)+\gamma (1-\gamma )\left(g_{22}\right(x\left)g_{22}\right(x\left)\frac{2}{6}\frac{1}{3}\right)]$
T 10=(((a,b),d),c)	$g_{22}\left(y\right)\left[{\gamma }^2\right(g_{32}\left(x\right)\frac{1}{3}\frac{1}{3}+g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})+{(1-\gamma )}^2(g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})$
	$+\gamma (1-\gamma )\left(g_{21}\right(x\left)g_{22}\right(x)\frac{1}{3}+g_{22}(x\left)g_{22}\right(x\left)\frac{1}{6}\frac{1}{3}\right)+\gamma (1-\gamma )\left(g_{22}\right(x\left)g_{22}\right(x\left)\frac{1}{6}\frac{1}{3}\right)]$
T 11=(b,(a,(c,d)))	$g_{22}\left(y\right)\left[{(1-\gamma )}^2\right(g_{32}\left(x\right)\frac{1}{3}\frac{1}{3}+g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})+{\gamma }^2(g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})$
	$+\gamma (1-\gamma )\left(g_{21}\right(x\left)g_{22}\right(x)\frac{1}{3}+g_{22}(x\left)g_{22}\right(x\left)\frac{1}{6}\frac{1}{3}\right)+\gamma (1-\gamma )\left(g_{22}\right(x\left)g_{22}\right(x\left)\frac{1}{6}\frac{1}{3}\right)]$
T 12=(((a,d),b),c)	$g_{22}\left(y\right)\left[{\gamma }^2\right(g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})+{(1-\gamma )}^2(g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})$
	$+\gamma (1-\gamma )\left(g_{22}\right(x\left)g_{22}\right(x\left)\frac{1}{6}\frac{1}{3}\right)+\gamma (1-\gamma )\left(g_{22}\right(x\left)g_{22}\right(x\left)\frac{1}{6}\frac{1}{3}\right)]$
T 13=(((b,d),a),c)	$g_{22}\left(y\right)\left[{\gamma }^2\right(g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})+{(1-\gamma )}^2(g_{32}\left(x\right)\frac{1}{3}\frac{1}{3}+g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})$
	$+\gamma (1-\gamma )\left(g_{22}\right(x\left)g_{22}\right(x\left)\frac{1}{6}\frac{1}{3}\right)+\gamma (1-\gamma )\left(g_{21}\right(x\left)g_{22}\right(x)\frac{1}{3}+g_{22}(x\left)g_{22}\right(x\left)\frac{1}{6}\frac{1}{3}\right)]$
T 14=(((a,c),d),b)	$g_{22}\left(y\right)\left[{(1-\gamma )}^2\right(g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})+{\gamma }^2(g_{32}\left(x\right)\frac{1}{3}\frac{1}{3}+g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})$
	$+\gamma (1-\gamma )\left(g_{22}\right(x\left)g_{22}\right(x\left)\frac{1}{6}\frac{1}{3}\right)+\gamma (1-\gamma )\left(g_{21}\right(x\left)g_{22}\right(x)\frac{1}{3}+g_{22}(x\left)g_{22}\right(x\left)\frac{1}{6}\frac{1}{3}\right)]$
T 15=(((a,d),c),b)	$g_{22}\left(y\right)\left[{\gamma }^2\right(g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})+{(1-\gamma )}^2(g_{33}\left(x\right)\frac{1}{6}\frac{1}{3})$
	$+\gamma (1-\gamma )\left(g_{22}\right(x\left)g_{22}\right(x\left)\frac{1}{6}\frac{1}{3}\right)+\gamma (1-\gamma )\left(g_{22}\right(x\left)g_{22}\right(x\left)\frac{1}{6}\frac{1}{3}\right)]$