Why doesn't current flow through an open branch? I know that current doesn't flow through open branch because current can't flow through air due to its high resistance .But i was thinking , what's the problem if current flows through an open wire (assumed 0 resistance for matter of circuit solving). I mean, isn't there a possibility that the charge flowing through the wire keeps accumulating at its end as it can't flow through air as battery is incapable of flowing it through the air. Why do we have to entirely abandon that wire?
 A: Current (or more properly, conduction current) can flow into an open wire and charge accumulate at the end IF there is displacement current to continue the circuit. Displacement current is not the flow of charge, but a change in the electric field. Two situations where this might happen is in an antenna, where the displacement current is provided by an electromagnetic wave,. The other situation is where the end of the wire is (in effect) one half of a capacitor, and the voltage across the capacitor is changing.
A: The answer is current actually does flow and charges do accumulate at ends. That's why arcs form in open circuits at high voltage. And as you mentioned the flow would be a momentary one and not continuous.
When we connect a battery to an open wire, the potential of the battery does push charges around. But as soon as the charge reaches the end and accumulate, they create an opposing potential and attain an equilibrium when the voltage induced at the open end is equal to the battery voltage.
Now, the voltage generated when $Q$ charge is put on an object is given by its Capacitance. So how much charge flows depends on the capacitance of the lose wire. Generally a bigger piece of wire can hold more charge at a given voltage. So if the lose wire is longer, more momentary currents will flow. ( apart from many other ways of explaining, this also explains why longer antennas capture better signals )
Also, the current flow will be momentary if the voltage is a constant voltage. If it is a varying voltage, the charge stored would also vary creating small
persistent currents.  example : line testers work even if we isolate ourselves from ground. This is because charge enters and exits our body due to varying voltage.
A: Voltage is the driving force .
If you have a real ( non ideal) wire and connect it across a battery, due to its non zero resistance, there will be a continuous drop in voltage. If the battery is 5 V rated, the potential at one end of the wire is 5, dropping down to 2.5 V at its mid length and all the way to zero at the other end.
Now if you take another wire and just touch one end at the mid way point of the above wire( and leave the other end hanging, not connected to anything. By this way, you get an open branch)  and wait for a second to let it achieve steady state, what happens?
The potential across the entire second wire is 2.5 V. Since current requires a potential difference , no current flows.
It is useful to think of charge as water flowing from the battery but it leads to confusions like the one you have ( i.e. does it get accumulated etc.)
However, your theory is not completely wrong since such a charge accumulation is exactly the reason a voltage is caused along a rod that is moving perpendicular to a magnetic field.
