Given that picture and your intended putting path, I would have expected the ball to break right all day. The peak of that rise is about double the radius of the branches of the tree nearest the pin, so the midpoint of your bump and run is to the right of that peak and so it'd break right. How much depends on the speed the ball is travelling, whether it's rolling or still bouncing at the midpoint, and other factors of the green.
Chad's answer has the two main things besides slope that determine the proper putting line for a shot; the grain of the grass, and the presence of any imperfections.
Grain is, quite simply, the direction in which the blades of grass grow. This depends on the species of grass, the location and orientation of the green relative to the path of the Sun, prevailing winds, and other factors (including the exact area of the green; it can change from one side of the green to the other). The best greens have no grain; the grass used grows straight up. But, most of those are high-maintenance grasses, and so most of your public and even semi-private courses, especially in southern climates where drought-tolerance is a plus, use something a little hardier that results in a grain. A putt directly against the grain will require more power than a putt with the grain, and putts will tend to break slightly in the direction of the grain, especially in the second half of their path as they slow to a stop, or when putting on a line within about thirty degrees of being directly with or against the grain.
In addition, greens that have been recently maintained often have rake furrows produced by the process of relevelling or thinning the green, which will give a more pronounced grain to the green surface.
Beyond the grain, the next most important thing is the softness or firmness of the ground under the grass, which is a primary determinant of the green's speed. The basic soil type, moisture content, and the amount of sand the groundskeeping staff have used to level it all contribute. The softer, the slower. The other primary determinant is overall grass height; most greens are kept close-mown, as short as 3/4". The taller, the slower, and the more pronounced the grain's effect.