Given some information for the length of piping, and components ( like Elbows, Tee's, Reducers etc..), and the conditions at A, namely pressure, flow rate, and the outlet conditions, then yeah, we can specify pressures at various locations in the system, and flow in each branch.
If the flow is steady (not varying in time) and the fluid is "virtually incompressible" (like water) then you can apply the following result between any two points in the system:
$$ \frac{P_1}{ \gamma} + z_1 + \frac{V_1^2}{2g} + h_p = \frac{P_2}{\gamma} + z_2 + \frac{V_2^2}{2g} + h_t + \sum_{1 \to 2} h_L $$
Each of the terms represents a form of "head", which basically an energy per unit mass of an element of flowing fluid.
## \frac{P}{\gamma} ## is a pressure head
## z ## is an elevation head
## \frac{V_1^2}{2g} ## is a kinetic energy head
## h_p ## is a pump head (external energy input)
## h_t ## is a turbine head (external energy withdrawal)
## \sum_{1 \to 2} h_L ## represents a thermal head that is created by viscous interaction ( friction) between the fluid and the pipe walls. Its always positive from the Second Law of Thermodynamics, and it is proportional to the kinetic energy head.
For the system you have shown this is going to amount to solving a system of non-linear equations resembling something like the following:
$$ \begin{array} \, \frac{P_A}{ \gamma } + \frac{V_A^2}{2g} = \frac{P_{A'} }{\gamma} + \frac{V_{A'}^2}{2g} + k_A ( Re, L , D) \frac{V_{A'}^2}{2g} \\ \frac{P_{A'} }{\gamma} + \frac{V_{A'}^2}{2g} = \frac{P_B}{\gamma} + \frac{V_B^2}{2g} + k_B( Re, L , D) \frac{V_B^2}{2g} \\ \frac{P_{A'} }{\gamma} + \frac{V_{A'}^2}{2g} = \frac{V_C^2}{2g} + k_C( Re, L , D) \frac{V_C^2}{2g} \\ Q_A = Q_B + Q_C \end{array} $$
The ##k( Re, L , D)## means it is a function of the Reynolds Number ( which has other dependencies: flow velocity, diameter, kinematic viscosity ), Length ## L ## and Diameter ## D## of pipe.
The system above is usually tackled by linearization and iteration until a desired accuracy for flow rates is achieved.
