# cemetery tree $\delta$

Let $$T_*$$ be a tree on which the parent $$e_*$$ of the root is added and $$x\in\mathbb{V}$$ a vertex in the Ulam-Harris tree. Then the tree $$T_*^{\leq x}$$ is defined as

$$\begin{cases} \mathsf{T}_*\setminus\{ v\in\mathbb{V}:x

If $$x\in\mathbb{V}$$, then $$T^{\leq x}_*$$ is the tree $$T_*$$ in which all strict descendants of $$v$$ are removed. Otherwise it is the cemetery tree $$\delta$$.

What is in this context a cemetery tree? Does it consists only of the root $$e$$?

arxiv.org/abs/1111.4313 [1] (page 4)

I ask for the definition because of Lemma 2.1 [1] on page 6: I want to check that the distributions of the backward tree $$\mathfrak{B}_x(\mathbb{T}_*)$$ and $$\mathbb{T}_*^{\leq \bar{x}}$$ are the same for $$x\notin\mathbb{T}_*$$.

I thought:

$$\mathfrak{B}_x(\mathbb{T}_*)\stackrel{(1)}{=}\Psi_x(\mathbb{T}_*^{\leq x})\stackrel{(2)}{=}\Psi_x(\delta)\stackrel{(3)}{=}\delta$$

(1) is the definition of the backward tree

(2) is because in the case $$x\notin\mathbb{T}_*$$: $$\mathbb{T}_*^{\leq x}=\delta$$

(3) is the definition of the function $$\Psi_x$$ in [1]

But then $$\delta$$ must be $$\mathbb{T}_*^{\leq \bar{x}}$$ which i do not see.

Do someone see my mistake?