of the lobe and the number of arcs on a given lobe represented by a white vertex $v$ is $\val(v)=|v|+1$. Hence the lengths of the arcs lying on the lobe represented by a vertex $v$ are in 1-1 correspondence with points of the simplex $|\Delta^{|v|}|$. The coordinates of $|\Delta^{|v|}|$ naturally correspond to the arcs of the lobe represented by $v$ on one hand and on the other hand in the dual b/w graph to the edges incident to $v$.

### The tree differential in the spineless case {#diffdef}

Let $\t\in \wlbptree$. We set $E_{angle}=E(\t)\setminus
(E_{leaf}(\t)\cup \{e_{root}\})$ and we denote by $\num_E:E_{angle} \rightarrow \{1,\dots,N\}$ the bijection which is induced by the linear order $\prec^{(\t,p)}$.

Let $\t\in \wlbptree$, $e\in E_{angle}$, $e=\{w,b\}$, with $w\in V_w$ and $b\in V_b$. Let $e-=\{w,b-\}$ be the edge preceding $e$ in the cyclic order $\prec^{\t}_w$ at $w$. Then $\del_e(\tau)$ is defined to be the planar tree obtained by collapsing the angle between the edge $e$ and its predecessor in the cyclic order of $w$ by identifying $b$ with $b-$ and $e$ with $e-$. Formally $w=\whitevert(e), e-=\prec^{\t}_w(e),\{b-\}= \del(e-)\cap
V_b(\t)$, $V_{\del_e(\tau)}=V(\t)/(b\sim b-)$, $E_{\del_e(\tau)}=E_{\tau}/(e\sim e-)$. The linear order of $\del_e(\t)$ is given by keeping the linear order at all vertices which 