Reversible and Irreversible Trees

A tree $\mathbb{T}=⟨T\le ⟩$ is reversible iff there is no order $\preccurlyeq ⊊\le$ such that $\mathbb{T}\cong ⟨T,\preccurlyeq ⟩$. Using a characterization of reversibility via back and forth systems we detect a wide class of non-reversible trees: bad trees" (having all branches of height ${\mathrm{ht}} ({\mathbb T})=|T|=|L_0|$, where $|T|$ is a regular cardinal). Consequently, a countable tree of height $ω$ and without maximal elements is reversible iff all its nodes are finite. We show that a tree ${\mathbb T}$ is non-reversible iff it contains a critical node" or an ``archetypical subtree" (parts of $\mathbb{T}$ with some combinatorial properties). In particular, a tree with finite nodes $\mathbb{T}$ is reversible iff it does not contain archetypical subtrees. Using that characterization we prove that if for each ordinal $\alpha \in [\omega ,\mathrm{h}\mathrm{t}(\mathbb{T}))$ all nodes of height $\alpha$ are of the same size, or the sequence $⟨⟨\mathrm{\mid }N\mathrm{\mid },\mathrm{\mid }N\uparrow \mathrm{\mid }⟩:\mathcal{N}(\mathbb{T})\ni N\subset L_{\alpha }⟩$ is finite-to-one, then $\mathbb{T}$ is reversible. Consequently, regular $n$-ary trees are reversible, reversible Aronszajn trees exist and, if there are Suslin or Kurepa trees, there are reversible ones. Also we show that for cardinals $\lambda >1$ and $\mu >0$ and ordinal $\alpha >0$ we have: the tree ${\bigcup }_{\mu }{}^{<\alpha }\lambda$ is reversible iff $\min \{\alpha ,\lambda \mu \}<\omega$.