ASCII art H trees

Charcoal, 22 bytes

Ｐ-²ＦＮＦ²«⟲Ｔ²+×⁺²κＸ²ι←‖Ｏ

Try it online! Link is to verbose version of code. 0-indexed. Explanation:

Ｐ-²

Print the initial three -s, leaving the cursor in the middle.

ＦＮ

Repeat for the number of times given.

Ｆ²«

Repeat twice for each H. Each loop creates a slightly bigger H from the previous loop, but we only want alternate Hs.

⟲Ｔ²

Rotate the figure.

+×⁺²κＸ²ι←

Draw half of the next line.

‖Ｏ

Reflect to complete the step.

The result at each iteration is as follows:

---

|   |
+---+
|   |

-+-   -+-
 |     | 
 +-----+ 
 |     | 
-+-   -+-

|   |   |   |
+-+-+   +-+-+
| | |   | | |
  |       |  
  +-------+  
  |       |  
| | |   | | |
+-+-+   +-+-+
|   |   |   |

-+-   -+-   -+-   -+-
 |     |     |     | 
 +--+--+     +--+--+ 
 |  |  |     |  |  | 
-+- | -+-   -+- | -+-
    |           |    
    +-----------+    
    |           |    
-+- | -+-   -+- | -+-
 |  |  |     |  |  | 
 +--+--+     +--+--+ 
 |     |     |     | 
-+-   -+-   -+-   -+-

Canvas, 20 19 bytes

ø⁸«╵［↷Ｌ⇵；ｌ⇵└┌├-×╋‼│

Try it here!

Explanation:

ø                    push an empty canvas
 ⁸«╵[              repeat input*2 + 1 times
     ↷               rotate clockwise
      L⇵             ceil(width/2)
        ;l⇵          ceil(height/2); leaves stack as [ ⌈½w⌉, canvas, ⌈½h⌉ ]
           └┌        reorder stack to [ canvas, ⌈½w⌉, ⌈½h⌉, ⌈½w⌉ ]
             ├       add 2 to the top ⌈w÷2⌉
              -×     "-" * (2 + ⌈w÷2⌉)
                ╋    in the canvas, at (⌈w÷2⌉; ⌈h÷2⌉) insert the dashes
                 ‼   normalize the canvas (the 0th iteration inserts at (0; 0) breaking things)
                  │  and palindromize horizontally