{The dendrite $P$}
\label{ex-dendrite-P}
In connection with the characterization of dendrites having
closed set of their end points (see
Property~\ref{ex-lccombs-clends} in~\ref{ex-lccombs} above)
the classes of those dendrites that do not contain a copy of
$F_\omega$ and those that do not contain a copy of the
\gs{comb $W$}{comb W} are
of some interest (see \cite[p. 8]{Arevalo+2001a}). The first
one is just the class of dendrites with finite orders of
ramification points, and it is known to have a universal
element according to Property~\ref{ex-other-universal-dendrites-univfinord}
in~\ref{ex-other-universal-dendrites}. The other class has
also a universal element $P$. Its construction, given in
\cite[p. 9]{Arevalo+2001a} is the following.

Let $p$ be any point in the plane $\mathbb R^2$ and let $p_1, p_2, \dots$ be
a sequence of points in $\mathbb R^2$ tending to $p$ and such that no three
of the points $p, p_1, p_2, \dots$ are collinear. Define $P_1 = \bigcup
\{\overline{pp_n}: n \in \mathbb N\}$. Then $P_1$ is homeomorphic to
$F_\omega$. For every $n, m \in \mathbb N$, let $p_{nm}$ be a point in
$\mathbb R^2 \setminus P_1$ such that for any fixed $n$ the sequence
$p_{n1}, p_{n2}, \dots$ tends to $p_n$, no three of the points $p_n, p_{n1},
p_{n2}, \dots$ are collinear, and that the continuum $P_2 = P_1 \cup \bigcup
\{\overline{p_np_{nm}}: n, m \in \mathbb N\}$ is a dendrite. We continue
constructing dendrites $P_3, P_4, \dots$ in the same manner, such that we
get an increasing sequence of dendrites $P_1 \subset P_2 \subset P_3
\subset \dots \subset P_n \subset P_{n+1} \subset \dots$ in $\mathbb R^2$
having the property that the closure of their union is also a dendrite.
Finally define
$$ P = \cl (\bigcup\{P_n: n \in \mathbb N\}). $$

The following properties of the
\es{dendrite $P$}{dendrite P}
are proved in \cite[Theorems 4.5 and 4.6, p.
10]{Arevalo+2001a}.

\rostere
\item A dendrite $X$ is homeomorphic to the dendrite $P$ if and only if
each of the following conditions is satisfied:
      \rostere
\item $X$ contains no copy of \gs{$W$}{comb W};
\item the set of all end points of $X$ is contained in the closure of
the set of all ramification points of $X$;
\item all ramification points of $X$ are of order $\omega$.
      \rosteree
\item The dendrite $P$ is \gs{universal}{universal dendrite} in the class
of all dendrites containing no copy of $W$.
\rosteree

See Figure A.

\FIGURE{A}{dendrite $P$}

\GIF{AA}{dendrite $P$ - an animation}
