Horizontal - Vertical connectors in a tree (in place of slanted lines)

Another option is to use forest. In this case, you can dispense with the phantom node and just use tier to align P7 with the P4, P5 etc. I've also updated your code to use the current syntax for arrows, provided by the arrows.meta library as this is now recommended. I like forest because specifying a tree involves much less typing!

Updated Solution (Forest version 2+)

\documentclass[tikz, border=5pt]{standalone}
\usepackage[edges]{forest}
\usetikzlibrary{arrows.meta}
\begin{document}

\tikzset{
  treenode/.style = {align=center, inner sep=2pt, rounded corners = 2pt, minimum width = 2cm, text centered, font=\sffamily},
  block/.style = {treenode, rectangle, white, font=\sffamily\bfseries, draw=black, fill=black},
}

% Forest version 2.1
\begin{forest}
  for tree={
    block,
    edge+={thick, -{Stealth[]}}
  },
  forked edges
  [P1
    [P2
      [P3
        [P4, tier=terminal]
        [P5]
      ]
    ]
    [P6, calign with current
      [P7, tier=terminal]
    ]
    [P8
      [P9
        [P10]
      ]
    ]
  ]
  ;
\end{forest}
\end{document}

Original Solution (Forest version 1)

\documentclass[tikz, border=5pt]{standalone}
\usepackage{forest}
\usetikzlibrary{arrows.meta}
\begin{document}

\tikzset{
  treenode/.style = {align=center, inner sep=2pt, rounded corners = 2pt, minimum width = 2cm, text centered, font=\sffamily},
  block/.style = {treenode, rectangle, white, font=\sffamily\bfseries, draw=black, fill=black},
}

\begin{forest}
  for tree={
    parent anchor=south,
    child anchor=north,
    block,
    edge path={
      \noexpand\path [-{Stealth[]}, \forestoption{edge}, thick]
        (!u.parent anchor) -- +(0,-5pt) -| (.child anchor)\forestoption{edge label};
    },
  }
  [P1
    [P2
      [P3
        [P4, tier=terminal]
        [P5]
      ]
    ]
    [P6, calign with current
      [P7, tier=terminal]
    ]
    [P8
      [P9
        [P10]
      ]
    ]
  ]
  ;
\end{forest}
\end{document}

Result

The output is the same, basically, regardless of which solution you use.

When you say horizontal/vertical, do you mean this?

Here is the code:

\documentclass[tikz,border=10]{standalone}
\usetikzlibrary{shapes,arrows}
\begin{document}
\tikzset{
  treenode/.style = {align=center, inner sep=2pt, rounded corners = 2pt, minimum width = 2cm, text centered, font=\sffamily},
  block/.style = {treenode, rectangle, white, font=\sffamily\bfseries, draw=black, fill=black},
  phantom/.style = {},
  edge from parent/.style={draw,red,thick},
  edge from parent path={(\tikzparentnode.south)-- ++(0,-3mm) -| (\tikzchildnode.north)}
}

\begin{tikzpicture}[->,>=stealth',level/.style={sibling distance = 3in/#1, level distance = 1.5cm}]
  \node [block] {P1}
    child {node [block] {P2}
      child {node [block] {P3}
        child {node [block] {P4}}
        child {node [block] {P5}}
      }
    }
    child {node [block] {P6}
      child {node [phantom] {}
        child {node [block] {P7}}
      }
    }
    child {node [block] {P8}
      child {node [block] {P9}
        child {node [block] {P10}}
      }
    }
  ;
\end{tikzpicture}
\end{document}

EDIT: I just discovered that we can use [edge from parent fork down] style from trees library to produce the same effect.

A PSTricks solution:

\documentclass{article}

\usepackage{pstricks}
\usepackage{xfp}

\psset{
  linecolor = red,
  arrows = ->
}

\def\Label(#1,#2)#3{%
  \psframe[
    framearc = 0.3,
    linecolor = black,
    fillstyle = solid,
    fillcolor = black
  ](\fpeval{#1-0.5*\widthBox},\fpeval{#2-0.5*\heightBox})%
   (\fpeval{#1+0.5*\widthBox},\fpeval{#2+0.5*\heightBox})
  \rput(#1,#2){\textcolor{white}{#3}}}

\begin{document}

% parameters
\def\widthBox{1.4}
\def\heightBox{0.4}
\def\horiDist{4.5}
\def\vertDist{1.6}

% drawing
\begin{pspicture}(\fpeval{2*\horiDist+1.75*\widthBox},\fpeval{4*\heightBox+3*\vertDist})
  % left
  \Label(\fpeval{1.25*\widthBox},\fpeval{2.5*\heightBox+2*\vertDist}){P2}
  \psline(\fpeval{1.25*\widthBox},\fpeval{2*\heightBox+2*\vertDist})%
         (\fpeval{1.25*\widthBox},\fpeval{2*\heightBox+\vertDist})
  \Label(\fpeval{1.25*\widthBox},\fpeval{1.5*\heightBox+\vertDist}){P3}
  \psline(\fpeval{1.25*\widthBox},\fpeval{\heightBox+\vertDist})%
         (\fpeval{1.25*\widthBox},\fpeval{\heightBox+0.5*\vertDist})%
         (\fpeval{0.5*\widthBox},\fpeval{\heightBox+0.5*\vertDist})%
         (\fpeval{0.5*\widthBox},\heightBox)
  \psline(\fpeval{1.25*\widthBox},\fpeval{\heightBox+0.5*\vertDist})%
         (\fpeval{2*\widthBox},\fpeval{\heightBox+0.5*\vertDist})%
         (\fpeval{2*\widthBox},\heightBox)
  \Label(\fpeval{0.5*\widthBox},\fpeval{0.5*\heightBox}){P4}
  \Label(\fpeval{2*\widthBox},\fpeval{0.5*\heightBox}){P5}
  % middle
  \Label(\fpeval{\horiDist+1.25*\widthBox},\fpeval{3.5*\heightBox+3*\vertDist}){P1}
  \psline(\fpeval{\horiDist+1.25*\widthBox},\fpeval{3*\heightBox+3*\vertDist})%
         (\fpeval{\horiDist+1.25*\widthBox},\fpeval{3*\heightBox+2*\vertDist})
  \Label(\fpeval{\horiDist+1.25*\widthBox},\fpeval{2.5*\heightBox+2*\vertDist}){P6}
  \psline(\fpeval{\horiDist+1.25*\widthBox},\fpeval{2*\heightBox+2*\vertDist})%
         (\fpeval{\horiDist+1.25*\widthBox},\heightBox)
  \Label(\fpeval{\horiDist+1.25*\widthBox},\fpeval{0.5*\heightBox}){P7}
  \psline(\fpeval{\horiDist+1.25*\widthBox},\fpeval{1.5*\heightBox+\vertDist-0.1})%
         (\fpeval{\horiDist+1.25*\widthBox},\heightBox)
  % right
  \Label(\fpeval{2*\horiDist+1.25*\widthBox},\fpeval{2.5*\heightBox+2*\vertDist}){P8}
  \psline(\fpeval{2*\horiDist+1.25*\widthBox},\fpeval{2*\heightBox+2*\vertDist})%
         (\fpeval{2*\horiDist+1.25*\widthBox},\fpeval{2*\heightBox+\vertDist})
  \Label(\fpeval{2*\horiDist+1.25*\widthBox},\fpeval{1.5*\heightBox+\vertDist}){P9}
  \psline(\fpeval{2*\horiDist+1.25*\widthBox},\fpeval{\heightBox+\vertDist})%
         (\fpeval{2*\horiDist+1.25*\widthBox},\heightBox)
  \Label(\fpeval{2*\horiDist+1.25*\widthBox},\fpeval{0.5*\heightBox}){P10}
  % connection
  \psline(\fpeval{\horiDist+1.25*\widthBox},\fpeval{3*\heightBox+2.5*\vertDist})%
         (\fpeval{1.25*\widthBox},\fpeval{3*\heightBox+2.5*\vertDist})%
         (\fpeval{1.25*\widthBox},\fpeval{3*\heightBox+2*\vertDist})
  \psline(\fpeval{\horiDist+1.25*\widthBox},\fpeval{3*\heightBox+2.5*\vertDist})%
         (\fpeval{2*\horiDist+1.25*\widthBox},\fpeval{3*\heightBox+2.5*\vertDist})%
         (\fpeval{2*\horiDist+1.25*\widthBox},\fpeval{3*\heightBox+2*\vertDist})
\end{pspicture}

\end{document}

With this code, all you have to do is choose the width (\widthBox) and the height (\heightBox) of the boxes, and the horizontal (\horiDist) and the vertical (\vertDist) distances between them. The drawing will then automatically be adjusted accordingly.