A simpler way to do this is to just use two trees and shift the second one (and change its branch properties) using TikZ's scope
mechanism:
\documentclass[10pt,letterpaper]{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{cancel}
\usepackage{tikz-qtree}
\begin{document}
\tikzset{edge from parent/.style=
{draw, edge from parent path={(\tikzparentnode) -- (\tikzchildnode)}}}
\begin{tikzpicture}[sibling distance=.5cm]
\Tree
[.n
[.$\dfrac{n}{2}$ [.$\dfrac{n}{4}$ ] [.$\dfrac{n}{4}$ ] [.$\dfrac{n}{4}$ ] ]
[.$\dfrac{n}{2}$ [.$\dfrac{n}{4}$ ] [.$\dfrac{n}{4}$ ] [.$\dfrac{n}{4}$ ] ]
[.$\dfrac{n}{2}$ [.$\dfrac{n}{4}$ ] [.$\dfrac{n}{4}$ ] [.$\dfrac{n}{4}$ ] ]
]
\begin{scope}[xshift=2.5in]
\tikzset{edge from parent/.style={edge from parent path={(\tikzparentnode) -- (\tikzchildnode)}},every tree node/.style={text width=4em,align=left}}
\Tree [.{$=n$}
[.{$= 3 \cdot \dfrac{n}{2}$}
[.{$= 3^2 \cdot \dfrac{n}{2^2}$} ]]]
\end{scope}
\end{tikzpicture}
\end{document}
I don't really get the question so I hope this is what you wanted. If you include a full document (such that we copy paste and see the problem on our systems) things are much more easier.
Here, you can change the default setting within a scope but your block
style had a node distance
which was resetting every time it is issued. I've made it 2mm such that we can see the difference easier.
\documentclass[tikz]{standalone}
\usetikzlibrary{arrows,shapes.geometric,positioning}
\begin{document}
\begin{tikzpicture}[decision/.style={diamond, draw, text width=4.5em, text badly centered, node distance=3.5cm, inner sep=0pt},
block/.style ={rectangle, draw, text width=6em, text centered, rounded corners, minimum height=4em, minimum height=2em},
cloud/.style ={draw, ellipse, minimum height=2em},
line/.style ={draw,-latex'},
node distance = 1cm,
auto]
\node [block] (1st) {1st};
\node [block, right= of 1st] (2nd1) {2nd1};
\begin{scope}[node distance=2mm and 10mm]%Here we change it for everything inside this scope
\node [block, above= of 2nd1] (2nd2) {2nd2};
\node [block, below= of 2nd1] (2nd3) {2nd3};
\node [block, right= of 2nd1] (3rd1) {3rd1};
\node [block, above= of 3rd1] (3rd2) {3rd2};
\node [block, above= of 3rd2] (3rd3) {3rd3};
\end{scope}
\node [block, below= of 3rd1] (3rd4) {3rd4};
\node [block, below= of 3rd4] (3rd5) {3rd5};
\path [line] (1st) -- (2nd1);
\path [line] (2nd1) -- (2nd2);
\path [line] (2nd1) -- (2nd3);
\path [line] (2nd2) -- (3rd3);
\path [line] (2nd1) -- (3rd1);
\path [line] (1st) -- (2nd1);
\end{tikzpicture}
\end{document}
Best Answer
You need to encourage
\missing
to expand at the right time: