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<v\}& \text{, falls } x\in\mathsf{T}_*\\ \delta &\text{, falls } x\notin\mathsf{T}_*.\\ \end{cases}

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?