Definition A tree is a partially ordered set $(T, <)$ such that for each $t \in T$, the set $\{s \in T : s < t\}$ is well-ordered by the relation $<$.
For trees $(T,<_T)$, $(S,<_S)$, $T\prec S$, if there is an embedding of $T$ into $S$, i.e. there exists an injection $f:T\rightarrow S$ such that for all $x,y\in T$, $x<_T y$ iff $f(x)<_S f(y)$.
Let $f_1,f_2$ witness that $T\prec S_1,S_2$ respectively. The "amalgam of $S_1,S_2$ over $T$" is a tree $R$ and embeddings $g_1,g_2$ such that $g_1,g_2$ witness that $S_1,S_2\prec R$ and $g_1\circ f_1=g_2\circ f_2$.
Question Does the amalgam always exist? Can we define it in many ways?
I was looking for a direct construction, i.e. defining $R$ from $T,S_1,S_2$, but I also found in Wikipedia the definition of a push-out. My background in category-theory is not strong enough. I could not tell if in the category of trees push-outs exists.
Any help either directly or through category-theory will be appreciated.
An amalgam does not always exist. Consider the poset $S=\{a,b,c,d,e\}$ with the ordering $a<c$, $a<e$, $b<c$, and $b<d$. Note that $S_1=S\setminus\{a\}$ and $S_2=S\setminus\{b\}$ are both trees with respect to this ordering. Setting $T=S_1\cap S_2$, I claim that there does not exist any amalgam of $S_1$ and $S_2$ over $T$ (with $f_1$ and $f_2$ the inclusion maps).
Indeed, suppose we had such an amalgam $R$; let us abuse notation and treat $S_1$ and $S_2$ as subsets of $R$. Then since $a,b<c$ we would have to have either $a\leq b$ or $b\leq a$ in $R$. But if $a\leq b$, then we would have $a\leq b<d$ in $R$, which is impossible since $a\not<d$ in $S_2$. Similarly, if $b\leq a$, we would have $b\leq a<e$ in $R$, which is impossible since $b\not<e$ in $S_1$.
Moreover, even when an amalgam does exist, and we require $R=g_1(S_1)\cup g_2(S_2)$ to avoid trivialities, the amalgam still will typically not be unique (up to isomorphism under $S_1$ and $S_2$). For instance, consider $S_1=\{a\}$, $S_2=\{b\}$, and $T=\emptyset$. Then there are four different minimal amalgams of $S_1$ and $S_2$ over $T$ (where "minimal" means $R=g_1(S_1)\cup g_2(S_2)$): one in which $a<b$, one in which $b<a$, one in which $a=b$, and one in which $a$ and $b$ are incomparable.
(For what it's worth, it's not hard to show that $R$ is a pushout of $S_1$ and $S_2$ over $T$ in the category of trees and embeddings iff $R$ is the unique minimal amalgam of $S_1$ and $S_2$. So the question of whether there exists an essentially unique amalgam is indeed equivalent to asking whether there is a pushout in this category.)