Quadratic Sequence tree branching node help

Question

I want to make something like this.

I tried using tree diagrams and got this code,

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[table]{xcolor}
\usepackage{mathtools}
\usepackage{amssymb}
\usepackage{enumitem}
\usepackage{multicol}
\usepackage{tasks}
\usepackage{geometry}
% \usepackage{poker}
\usepackage{array}
\usepackage{indentfirst}
\usepackage{tikz}
\usepackage{tkz-euclide}
\usetikzlibrary{arrows}
\usetkzobj{all}
\usetikzlibrary{trees}
 \geometry{
 a4paper,
 total={170mm,257mm},
 left=20mm,
 top=20mm,
 bottom=20mm
 }
 \settasks{
  style = enumerate ,
  counter-format = tsk),
  label-align = right,
  label-width = 2em,
  label-offset = .75em
  %label-format = \bfseries
}
\newcolumntype{L}[1]{>{\raggedright\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
\newcommand{\N}{\mathbb N}
\newcommand{\Q}{\mathbb Q}
\newcommand{\C}{\mathbb C}
\newcommand{\Z}{\mathbb Z}
\setcounter{secnumdepth}{0}
\begin{document}

% Set the overall layout of the tree
\tikzstyle{level 2}=[level distance=2cm, sibling distance=6cm]
\tikzstyle{level 3}=[level distance=2cm, sibling distance=3cm]

% Define styles for bags and leafs
\tikzstyle{bag} = [text width=4em, text centered]
\tikzstyle{end} = [circle, minimum width=3pt,fill, inner sep=0pt]

\begin{tikzpicture}[grow=up, sloped]
\node {}
    child {
        node[bag] {$2a$}
        child {
            node[bag] {$5a+b$}
            child {
                node {$16a+4b+c$}
            }
            child {
                node {$9a+3b+c$}
            }
        }
        child {
            node[bag] {$3a+b$}
            child {
                node {$4a+2b+c$}
            }
            child {
                node {$a+b+c+d$}
            }
        }
    };
\end{tikzpicture}
\end{document}

which yields this result:

The problem with this is if I do another set that includes another 2a, those two 2a's will branch into one node. How should I change my code?

Answer 1

Trees have a single root and every non-root node has one parent. Generally, packages designed to draw trees are specialised software for drawing trees in this sense and not, say, more generalised graphs. The TikZ library doesn't mind multiple roots, but it is still tricky to deal with multiple parents and so on.

I think it would be easier to think of the diagram as a matrix with some edges drawn between some of the nodes.

For example,

\documentclass[border=10pt]{standalone}
\usepackage{tikz,amsmath}
\usetikzlibrary{matrix}
\begin{document}
\begin{tikzpicture}
  \matrix (M) [matrix of math nodes]
  {
    & n=1 && n=2 && n=3 && n=4\\
   T_n & a+b+c && 4a+2b+c && 9a+3b+c && 16a+4b+c\\
   \text{1\textsuperscript{st} difference} & & 3a+b && 5a+b && 7a+b & \\
   \text{2\textsuperscript{nd} difference} & & & 2a && 2a &&\\
  };
  \draw (M-2-2) -- (M-3-3) -- (M-4-4) -- (M-3-5) edge (M-2-4) edge (M-2-6) -- (M-4-6) -- (M-3-7) edge (M-2-6) -- (M-2-8) ;
\end{tikzpicture}

\end{document}