# Specifying the depth of a binary tree on the side

I'm trying to draw a tree, and I want to specify its depth along the side with a double headed arrow with the depth written on the arrow. Here is what I have so far:

\begin{tikzpicture}[<-, >=stealth', auto, semithick, level/.style={sibling distance=60mm/#1}]
%\node  {} child {
\node [circle,draw] (z){$\land$}
child {node [circle,draw] (a) {$\lor$}
child {node [circle,draw] (b) {$\land$}
child {node {$\vdots$}
child {node [rectangle,draw] (d) {$x_1$}}
child {node [rectangle,draw] (e) {$x_2$}}
}
child {node {$\vdots$}}
}
child {node [circle,draw] (g) {$\land$}
child {node {$\vdots$}}
child {node {$\vdots$}}
}
}
child {node [circle,draw] (j) {$\lor$}
child {node [circle,draw] (k) {$\land$}
child {node {$\vdots$}}
child {node {$\vdots$}}
}
child {node [circle,draw] (l) {$\land$}
child {node {$\vdots$}}
child {node (c){$\vdots$}
child {node [rectangle,draw] (o) {$x_{n-1}$}}
child {node [rectangle,draw] (p) {$x_n$}
%child [grow=right] {node (q) {$=$} edge from parent[draw=none]
%child [grow=right] {node (q) {$O_{k = \lg n}(n)$} edge from parent[draw=none]
%child [grow=up] {node (r) {$\vdots$} edge from parent[draw=none]
% child [grow=up] {node (s) {$O_2(n)$} edge from parent[draw=none]
% child [grow=up] {node (t) {$O_1(n)$} edge from parent[draw=none]
%  child [grow=up] {node (u) {$O_0(n)$} edge from parent[draw=none]}
% }
% }
%}
% child [grow=down] {node (v) {$O(n \cdot \lg n)$}edge from parent[draw=none]}
%}
%}
}
}
}
};

\path (o) -- (e) node (x) [midway] {$\cdots$}
child [grow=down] {
node (y) {}%{$O\left(\displaystyle\sum_{i = 0}^k 2^i \cdot \frac{n}{2^i}\right)$}
edge from parent[draw=none]
};
node (w) {};%[midway] {$O\left(\displaystyle\sum_{i = 0}^k n\right) = O(k \cdot n)$};
% \path (q) -- (v) node [midway] {=};
% \path (e) -- (x) node [midway] {+};
% \path (o) -- (x) node [midway] {+};
% \path (y) -- (w) node [midway] {$=$};
% \path (v) -- (w) node [midway] {$\Leftrightarrow$};
%\path (r) -- (c) node [midway] {$\cdots$};
\end{tikzpicture}
\end{preview}

I'm trying to modify the code from here. How can I fix this problem?

(Unrelated: Is there a way I can have an arrow coming out of the root node as well?)

-

Hope this proposed solution understands the OP well. Here an double head arrow is defined via \pgfarrowsdeclarecombine

Code

\documentclass[border=20pt]{standalone}%
\usepackage{tikz}
\usetikzlibrary{shapes,calc,positioning,arrows,}

{stealth'}{stealth'}{stealth'}{stealth'}

\begin{document}

\begin{tikzpicture}[<-, >=stealth', auto, semithick,
level/.style={sibling distance=60mm/#1}
]

%\node  {} child {
\node [circle,draw] (z){$\land$}
child {node [circle,draw] (a) {$\lor$}
child {node [circle,draw] (b) {$\land$}
child {node {$\vdots$}
child {node [rectangle,draw] (d) {$x_1$}}
child {node [rectangle,draw] (e) {$x_2$}}
}
child {node {$\vdots$}}
}
child {node [circle,draw] (g) {$\land$}
child {node {$\vdots$}}
child {node {$\vdots$}}
}
}
child {node [circle,draw] (j) {$\lor$}
child {node [circle,draw] (k) {$\land$}
child {node {$\vdots$}}
child {node {$\vdots$}}
}
child {node [circle,draw] (l) {$\land$}
child {node {$\vdots$}}
child {node (c){$\vdots$}
child {node [rectangle,draw] (o) {$x_{n-1}$}}
child {node [rectangle,draw] (p) {$x_n$}
}
}
}
};

\path (o) -- (e) node (x) [midway] {$\cdots$}
child [grow=down] {
edge from parent[draw=none]
};
\draw [twotriang-twotriang] ($(7cm,0)+(z.north)$)  coordinate(P)--node[midway]{\rotatebox[origin=c]{270}{depth}} (P|-p.south);
\draw [->] (z) -- ++(0,1cm);
\end{tikzpicture}

\end{document}
-