YES
We show the termination of the TRS R:

  app(app(mapbt(),f),app(leaf(),x)) -> app(leaf(),app(f,x))
  app(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app(app(app(branch(),app(f,x)),app(app(mapbt(),f),l)),app(app(mapbt(),f),r))

Instead we prove termination of the uncurried system:

  app(branch_0(),y) -> branch_1(y)
  app(branch_1(x1),y) -> branch_2(x1,y)
  app(branch_2(x1,x2),y) -> branch_3(x1,x2,y)
  app(leaf_0(),y) -> leaf_1(y)
  app(mapbt_0(),y) -> mapbt_1(y)
  app(mapbt_1(x1),y) -> mapbt_2(x1,y)
  mapbt_2(f,leaf_1(x)) -> leaf_1(app(f,x))
  mapbt_2(f,branch_3(x,l,r)) -> branch_3(app(f,x),mapbt_2(f,l),mapbt_2(f,r))


-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: app#(mapbt_1(x1),y) -> mapbt_2#(x1,y)
p2: mapbt_2#(f,leaf_1(x)) -> app#(f,x)
p3: mapbt_2#(f,branch_3(x,l,r)) -> app#(f,x)
p4: mapbt_2#(f,branch_3(x,l,r)) -> mapbt_2#(f,l)
p5: mapbt_2#(f,branch_3(x,l,r)) -> mapbt_2#(f,r)

and R consists of:

r1: app(branch_0(),y) -> branch_1(y)
r2: app(branch_1(x1),y) -> branch_2(x1,y)
r3: app(branch_2(x1,x2),y) -> branch_3(x1,x2,y)
r4: app(leaf_0(),y) -> leaf_1(y)
r5: app(mapbt_0(),y) -> mapbt_1(y)
r6: app(mapbt_1(x1),y) -> mapbt_2(x1,y)
r7: mapbt_2(f,leaf_1(x)) -> leaf_1(app(f,x))
r8: mapbt_2(f,branch_3(x,l,r)) -> branch_3(app(f,x),mapbt_2(f,l),mapbt_2(f,r))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: app#(mapbt_1(x1),y) -> mapbt_2#(x1,y)
p2: mapbt_2#(f,branch_3(x,l,r)) -> mapbt_2#(f,r)
p3: mapbt_2#(f,branch_3(x,l,r)) -> mapbt_2#(f,l)
p4: mapbt_2#(f,branch_3(x,l,r)) -> app#(f,x)
p5: mapbt_2#(f,leaf_1(x)) -> app#(f,x)

and R consists of:

r1: app(branch_0(),y) -> branch_1(y)
r2: app(branch_1(x1),y) -> branch_2(x1,y)
r3: app(branch_2(x1,x2),y) -> branch_3(x1,x2,y)
r4: app(leaf_0(),y) -> leaf_1(y)
r5: app(mapbt_0(),y) -> mapbt_1(y)
r6: app(mapbt_1(x1),y) -> mapbt_2(x1,y)
r7: mapbt_2(f,leaf_1(x)) -> leaf_1(app(f,x))
r8: mapbt_2(f,branch_3(x,l,r)) -> branch_3(app(f,x),mapbt_2(f,l),mapbt_2(f,r))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{17, x1 - 2}
        mapbt_1_A(x1) = max{21, x1 + 2}
        mapbt_2#_A(x1,x2) = max{18, x1 - 1}
        branch_3_A(x1,x2,x3) = max{x1 - 42, x2 - 53, x3 + 6}
        leaf_1_A(x1) = max{16, x1 + 4}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            app#_A(x1,x2) = x2
            mapbt_1_A(x1) = x1 + 3
            mapbt_2#_A(x1,x2) = x2
            branch_3_A(x1,x2,x3) = x1 + x2 + x3 + 7
            leaf_1_A(x1) = x1 + 11
            app#!_A(x1,x2) = x1 + 12
            mapbt_1!_A(x1) = 14
            mapbt_2#!_A(x1,x2) = x2 + 4
            branch_3!_A(x1,x2,x3) = x1 + x2 + x3 + 3
            leaf_1!_A(x1) = x1 + 13
  
    partial status:
  
      pi(app#) = []
      pi(mapbt_1) = [1]
      pi(mapbt_2#) = []
      pi(branch_3) = [3]
      pi(leaf_1) = [1]

The following rules are removed:

  p1, p2, p3, p4, p5

Then no dependency pair remains.