Presenting a {forest} tree from bottom to top in beamer

You can use visible on style defined by Qrrbrbirlbel but, instead of using it for tree, do it for ancestors'. for ancestors' style is applied to actual node and parents, therefore visible on=<1-> means this node and parents and edges between them will be visible from slide 1, but applying visible on=<2-> to parent nodes they are not visible until slide 2.

Edges between nodes and parents are also visible, but I don't know how to change slide value in style definition. The solution I can provide to show edges only when ancestors are shown is manually fix edges visibility with

[{[ajout]}\\\textit{dans la cuisine}, align=center, visible on=<2->,edge={/tikz/visible on=<3->}]

Here is the complete code

\documentclass{beamer}
\usepackage{forest}

\setbeamertemplate{navigation symbols}{} %Remove navigation bar
\tikzset{
    invisible/.style={opacity=0,text opacity=0},
    visible on/.style={alt=#1{}{invisible}},
    alt/.code args={<#1>#2#3}{%
      \alt<#1>{\pgfkeysalso{#2}}{\pgfkeysalso{#3}} % \pgfkeysalso doesn't change the path
    },
}
\forestset{
  visible on/.style={
    for ancestors'={
      /tikz/visible on={#1},
      edge={/tikz/visible on={#1}}}}}

\begin{document}
\begin{frame}[plain]{Same with forest}
  \begin{forest}
    [VP, calign=last, visible on=<4->
      [{[ajout]}\\\textit{rapidement}, 
      align=center,
      base=top, visible on=<3->, ]
      [VP, calign=first, visible on=<3->, 
        [VP, calign=first, visible on=<2->, 
          [V\\\textit{mange}, align=center,base=top, visible on=<1->, ]
          [{[compl]}\\\textit{une orange}, align=center,base=top, visible on=<1->, ]
        ]
        [{[ajout]}\\\textit{dans la cuisine}, align=center, visible on=<2->,]
      ]
    ]
  \end{forest}

\end{frame}
\end{document}

Note: since forest v2.0, key for ancestors' should be changed to for current and ancestors.

Here's a variation of Ignasi's answer which provides a show up tree style for revealing trees incrementally upwards. This style also illustrates a way to get the edges to appear with their parents when passing a single overlay specification, rather than with their children, which Ignasi suggested handling by passing two different overlay specifications.

\documentclass{beamer}
\usepackage{forest}

\setbeamertemplate{navigation symbols}{} %Remove navigation bar
\tikzset{
  invisible/.style={opacity=0,text opacity=0},
  visible on/.style={alt=#1{}{invisible}},
  alt/.code args={<#1>#2#3}{%
    \alt<#1>{\pgfkeysalso{#2}}{\pgfkeysalso{#3}} % \pgfkeysalso doesn't change the path
  },
}
\forestset{
  show up tree/.style={
    before typesetting nodes={
      tempcounta/.max={level}{tree},
      tempcounta'+=1,
      for tree={
        /tikz/visible on/.process={ ROw2+nw {tempcounta}{level}{##1-##2}{<##1->} },
        edge+/.process={ ROw2+nw {tempcounta}{level}{(##1-##2)+1} {/tikz/visible on={ {<##1->} } }},
      }
    }
  }
}

\begin{document}
\begin{frame}[plain]{Same with forest}
  \begin{forest}
    show up tree
    [VP, calign=last, 
      [{[ajout]}\\\textit{rapidement}, 
      align=center,
      base=top,  ]
      [VP, calign=first,  
        [VP, calign=first,  
          [V\\\textit{mange}, align=center,base=top,  ]
          [{[compl]}\\\textit{une orange}, align=center,base=top,  ]
        ]
        [{[ajout]}\\\textit{dans la cuisine}, align=center, ]
      ]
    ]
  \end{forest}

\end{frame}
\end{document}