Let $κ$ be an uncountable regular cardinal and $T$ be a tree of height $\kappa$.
(1) A function $r : T → T$ is called regressive if $r(t) <_T t$ holds for every $t \in T$ \ {root($T$)}.
(2) The tree $T$ is called special if there is a regressive map $r : T → T$ with the property that
$r^{-1}$[{$t$}] is the union of less than $κ$-many antichains in $T$, for every $t \in T$.
In other words, for every $t ∈ T$ there is some $λ < κ$ and a function $c_t:$$r^{-1}$[{$t$}]$\to \lambda$ such that
$c_t(s_0) \neq c_t(s_1)$ for all $s_o,s_1 \in T$ with $r(s_0)=r(s_1)=t$ and $s_0<_Ts_1$.
What I would like to show is, that for any uncountable, regular cardinal $\kappa$ and given $T$ a tree of height $\kappa$, then if $T$ is special, $T$ cannot have cofinal branches.
Jech's book is giving this claim as to be straightforward, but still can someone give a bit more precise explanation why this holds.
Suppose there is a cofinal branch, namely a chain $\{t_\alpha\mid\alpha<\kappa\}$ with $t_\alpha$ in the $\alpha$th level of $T$.
Fix your regressive function, $r$, and consider $f(\alpha)=\beta\iff r(t_\alpha)=t_\beta$ when restricting $r$ to the branch. By Fodor's lemma, there is a stationary subset $S\subseteq\kappa$ such that $f\restriction S$ is constant with value $\beta$. But that means that $r^{-1}[\{t_\beta\}]$ is not the union of $\lambda<\kappa$ antichains since it contains a cofinal subset of the branch (and therefore has to be written as the union of $\kappa$ antichains).