Draw a Fibonacci Network!
Pyth, 31 bytes

Ms+0m+Wd+J+G"| "+G1+\-gJtd_UHgb


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How it works

M                                  def g(G, H):
    m                     _UH        map for d in [H - 1, …, 0]:
         J+G"| "                       assign J = G + "| "
        +       +G1                    J + G + 1
     +Wd                               conditional +, if d is truthy (nonzero):
                   +\-gJtd               "-" + g(J, d - 1)
  +0                                 prepend 0
 s                                   concatenate
                             gbQ   g("\n", input)


Output for 6

0
| 
1-0
| | 
| 1-0
| | | 
| | 1-0
| | | 
| | 1
| | 
| 1-0
| | | 
| | 1
| | 
| 1-0
| | 
| 1
| 
1-0
| | 
| 1-0
| | | 
| | 1
| | 
| 1-0
| | 
| 1
| 
1-0
| | 
| 1-0
| | 
| 1
| 
1-0
| | 
| 1
| 
1-0
| 
1


Pyth, 43 bytes

Mc+WH`G++WG\-+j"
| "g0tH"
`-"j"
  "g1tHbjg0


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Output for 6

0-1-0-1-0-1-0
  |   |   `-1
  |   `-1-0-1
  |     `-1-0
  |       `-1
  `-1-0-1-0-1
    |   `-1-0
    |     `-1
    `-1-0-1-0
      |   `-1
      `-1-0-1
        `-1-0
          `-1