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In the book Tomb of Battle, initiator classes gain maneuvers which are in a sense, comparable to a spellcaster's spells. But while an initiator must qualify with many lesser examples of a certain discipline before obtaining a high level example of the school, a spellcaster has no such limitation. This variant represents a minor nerf to a spellcaster's power in the form of requiring certain chains of spells from the same school, in order to obtain the higher level versions. This is both thematic, showing that one must have a mastery over, say, lesser transmutation, before one may polymorph into dragons at a thought, and provides a coherent theme to the caster as a whole.
Cantrips and 1st level spells require no pre-requisites and may me taken as one pleases. However from level 2 and onward, spells gain a pre-resquisite of other spells known in that spell school equal to half the spell level (rounded down). For example, to learn scorching ray (a level 2 evocation) you need to know at least 1 evocation from any other level. Magic Missile would do just fine, allowing you to learn scorching ray. To learn an 8th or 9th level spell, you need to have 4 spells in that spell school known. And so forth. Universal spells may be the exception allowing any spells to qualify, or have different pre-requisites for the few spells which exist.
As a variant on the variant, a DM may add additional restrictions if needed, such as requiring knowledge of both 1 evocation and 1 fire spell (in which case, Burning Hands fulfills both requirements), or additional restrictions, especially on problematic spells.
This works well against wizards, but may hurt sorcerers and their limited spells known the worst. Consider either applying this only to prepared casters, or granting spontaneous casters either extra spells known, or an easier means of obtaining new spells if it is too hard on spontaneous casters. This variant can also work for casters who have all their spells known (such as clerics), by requiring that the prepared spells for the day must qualify, instead of spells known.