diff --git a/stdlib/LinearAlgebra/src/structuredbroadcast.jl b/stdlib/LinearAlgebra/src/structuredbroadcast.jl
index 3b5ba2475aeb1..a665e21731752 100644
--- a/stdlib/LinearAlgebra/src/structuredbroadcast.jl
+++ b/stdlib/LinearAlgebra/src/structuredbroadcast.jl
@@ -60,13 +60,21 @@ structured_broadcast_alloc(bc, ::Type{<:Diagonal}, ::Type{ElType}, n) where {ElT
 # Bidiagonal is tricky as we need to know if it's upper or lower. The promotion
 # system will return Tridiagonal when there's more than one Bidiagonal, but when
 # there's only one, we need to make figure out upper or lower
-find_bidiagonal() = throw(ArgumentError("could not find Bidiagonal within broadcast expression"))
-find_bidiagonal(a::Bidiagonal, rest...) = a
-find_bidiagonal(bc::Broadcast.Broadcasted, rest...) = find_bidiagonal(find_bidiagonal(bc.args...), rest...)
-find_bidiagonal(x, rest...) = find_bidiagonal(rest...)
+merge_uplos(::Nothing, ::Nothing) = nothing
+merge_uplos(a, ::Nothing) = a
+merge_uplos(::Nothing, b) = b
+merge_uplos(a, b) = a == b ? a : 'T'
+
+find_uplo(a::Bidiagonal) = a.uplo
+find_uplo(a) = nothing
+find_uplo(bc::Broadcasted) = mapreduce(find_uplo, merge_uplos, bc.args, init=nothing)
+
 function structured_broadcast_alloc(bc, ::Type{<:Bidiagonal}, ::Type{ElType}, n) where {ElType}
-    ex = find_bidiagonal(bc)
-    return Bidiagonal(Array{ElType}(undef, n),Array{ElType}(undef, n-1), ex.uplo)
+    uplo = find_uplo(bc)
+    if uplo == 'T'
+        return Tridiagonal(Array{ElType}(undef, n-1), Array{ElType}(undef, n), Array{ElType}(undef, n-1))
+    end
+    return Bidiagonal(Array{ElType}(undef, n),Array{ElType}(undef, n-1), uplo)
 end
 structured_broadcast_alloc(bc, ::Type{<:SymTridiagonal}, ::Type{ElType}, n) where {ElType} =
     SymTridiagonal(Array{ElType}(undef, n),Array{ElType}(undef, n-1))
diff --git a/stdlib/LinearAlgebra/test/structuredbroadcast.jl b/stdlib/LinearAlgebra/test/structuredbroadcast.jl
index 72d304ac6da3b..b8f5e97311588 100644
--- a/stdlib/LinearAlgebra/test/structuredbroadcast.jl
+++ b/stdlib/LinearAlgebra/test/structuredbroadcast.jl
@@ -160,4 +160,49 @@ end
     @test L .+ UnitL .+ UnitU .+ U .+ D == L + UnitL + UnitU + U + D
     @test L .+ U .+ D .+ D .+ D .+ D == L + U + D + D + D + D
 end
+@testset "Broadcast Returned Types" begin
+    # Issue 35245
+    N = 3
+    dV = rand(N)
+    evu = rand(N-1)
+    evl = rand(N-1)
+
+    Bu = Bidiagonal(dV, evu, :U)
+    Bl = Bidiagonal(dV, evl, :L)
+    T = Tridiagonal(evl, dV * 2, evu)
+
+    @test typeof(Bu .+ Bl) <: Tridiagonal
+    @test typeof(Bl .+ Bu) <: Tridiagonal
+    @test typeof(Bu .+ Bu) <: Bidiagonal
+    @test typeof(Bl .+ Bl) <: Bidiagonal
+    @test Bu .+ Bl == T
+    @test Bl .+ Bu == T
+    @test Bu .+ Bu == Bidiagonal(dV * 2, evu * 2, :U)
+    @test Bl .+ Bl == Bidiagonal(dV * 2, evl * 2, :L)
+
+
+    @test typeof(Bu .* Bl) <: Tridiagonal
+    @test typeof(Bl .* Bu) <: Tridiagonal
+    @test typeof(Bu .* Bu) <: Bidiagonal
+    @test typeof(Bl .* Bl) <: Bidiagonal
+
+    @test Bu .* Bl == Tridiagonal(zeros(N-1), dV .* dV, zeros(N-1))
+    @test Bl .* Bu == Tridiagonal(zeros(N-1), dV .* dV, zeros(N-1))
+    @test Bu .* Bu == Bidiagonal(dV .* dV, evu .* evu, :U)
+    @test Bl .* Bl == Bidiagonal(dV .* dV, evl .* evl, :L)
+
+    Bu2 =  Bu .* 2
+    @test typeof(Bu2) <: Bidiagonal && Bu2.uplo == 'U'
+    Bu2 = 2 .* Bu
+    @test typeof(Bu2) <: Bidiagonal && Bu2.uplo == 'U'
+    Bl2 =  Bl .* 2
+    @test typeof(Bl2) <: Bidiagonal && Bl2.uplo == 'L'
+    Bu2 = 2 .* Bl
+    @test typeof(Bl2) <: Bidiagonal && Bl2.uplo == 'L'
+
+    # Example of Nested Brodacasts
+    tmp = (1 .* 2) .* (Bidiagonal(1:3, 1:2, 'U') .* (3 .* 4)) .* (5 .* Bidiagonal(1:3, 1:2, 'L'))
+    @test typeof(tmp) <: Tridiagonal
+
+end
 end